{"id":494200,"date":"2026-09-10T09:10:14","date_gmt":"2026-09-10T09:10:14","guid":{"rendered":"https:\/\/savepearlharbor.com\/?p=494200"},"modified":"-0001-11-30T00:00:00","modified_gmt":"-0001-11-29T21:00:00","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=494200","title":{"rendered":"\u041f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0441\u043e \u0432\u0441\u0435\u0445 \u0441\u0442\u043e\u0440\u043e\u043d"},"content":{"rendered":"<div xmlns=\"http:\/\/www.w3.org\/1999\/xhtml\">\n<p>\u0423\u0447\u0430\u0441\u044c \u0432 \u0438\u043d\u0441\u0442\u0438\u0442\u0443\u0442\u0435, \u044f \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u043b \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0432 \u0441\u043e\u0432\u0435\u0440\u0448\u0435\u043d\u043d\u043e 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src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/942\/9cb\/2ef\/9429cb2ef6c8a7228d45d00434956476.png\" width=\"1392\" height=\"633\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/942\/9cb\/2ef\/9429cb2ef6c8a7228d45d00434956476.png 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/942\/9cb\/2ef\/9429cb2ef6c8a7228d45d00434956476.png 781w\" loading=\"lazy\" decode=\"async\"\/><\/figure>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"x^{2}\" alt=\"x^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c66\/c66452631491acdbf8e5ed69dfd19681.svg\" width=\"18\" height=\"16\" data-width=\"2.282\" data-height=\"2.025\" data-vertical-align=\"-0.025\" style=\"width: 2.282ex; height: 2.025ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" 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decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\left( y,f(y) \\right)\" alt=\"\\left( y,f(y) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/59\/590\/590c9a4a25f3992362a55451b6531078.svg\" width=\"64\" height=\"18\" data-width=\"7.988\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.988ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/59\/590\/590c9a4a25f3992362a55451b6531078.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/59\/590\/590c9a4a25f3992362a55451b6531078.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043c\u044b \u0432\u0441\u0435\u0433\u0434\u0430 \u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u043c\u0441\u044f \u043d\u0430\u0434 \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u043c. \u042d\u0442\u043e \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e\u043c \u0419\u0435\u043d\u0441\u0435\u043d\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f \\left( tx + (1 - t)y \\right) \\leq t \\cdot f(x) + (1 - t) \\cdot f(y)\\ \\forall x,y\\ \\forall t \\in \\lbrack 0,1\\rbrack\" alt=\"f \\left( tx + (1 - t)y \\right) \\leq t \\cdot f(x) + (1 - t) \\cdot f(y)\\ \\forall x,y\\ \\forall t \\in \\lbrack 0,1\\rbrack\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b5\/b58\/b58f314242777a3ba414dd3f7ccddc32.svg\" width=\"446\" height=\"18\" data-width=\"55.76\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 55.76ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b5\/b58\/b58f314242777a3ba414dd3f7ccddc32.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b5\/b58\/b58f314242777a3ba414dd3f7ccddc32.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u043e-\u0432\u0442\u043e\u0440\u044b\u0445, \u0434\u043b\u044f \u0433\u043b\u0430\u0434\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u044c \u043c\u043e\u0436\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c \u0442\u0430\u043a: \u0433\u0440\u0430\u0444\u0438\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043b\u0435\u0436\u0438\u0442 \u043d\u0430\u0434 \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u043c\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(y) \\geq f(x) + f^{'}(x)(y - x)\\ \\ \\forall x,y\" alt=\"f(y) \\geq f(x) + f^{'}(x)(y - x)\\ \\ \\forall x,y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/21c\/21cec82710abdb3e7822734da31b6e2c.svg\" width=\"256\" height=\"22\" data-width=\"31.971\" data-height=\"2.714\" data-vertical-align=\"-0.791\" style=\"width: 31.971ex; 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 https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4e\/4e2\/4e21530ae4646555fd5a7542747ef980.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u043c\u0441\u044f \u043d\u0430 (2). \u0417\u0430\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"f\" alt=\"f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8f\/8fa\/8fa14cdd754f91cc6554c9e71929cce7.svg\" width=\"12\" height=\"16\" data-width=\"1.244\" data-height=\"2.059\" data-vertical-align=\"-0.464\" style=\"width: 1.244ex; height: 2.059ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8f\/8fa\/8fa14cdd754f91cc6554c9e71929cce7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8f\/8fa\/8fa14cdd754f91cc6554c9e71929cce7.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043c\u043e\u0436\u043d\u043e \u043e\u043f\u0438\u0441\u0430\u0442\u044c \u0434\u0432\u0443\u043c\u044f \u0441\u043f\u043e\u0441\u043e\u0431\u0430\u043c\u0438:<\/p>\n<ul>\n<li>\n<p>\u043d\u0430\u0431\u043e\u0440\u043e\u043c \u0432\u0441\u0435\u0445 \u0442\u043e\u0447\u0435\u043a <img decoding=\"async\" class=\"formula inline\" source=\"(x,y)\" alt=\"(x,y)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/90c\/90cbc22edf225adf8a68974f51227f05.svg\" width=\"41\" height=\"18\" data-width=\"5.169\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.169ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/90c\/90cbc22edf225adf8a68974f51227f05.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/90c\/90cbc22edf225adf8a68974f51227f05.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442\u0430\u043a\u0438\u0445 \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"y = f(x)\" alt=\"y = f(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fd9\/fd91c508f91c2c84498680bd337c1d7a.svg\" width=\"67\" height=\"18\" data-width=\"8.424\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.424ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fd9\/fd91c508f91c2c84498680bd337c1d7a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fd9\/fd91c508f91c2c84498680bd337c1d7a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p>\u043d\u0430\u0431\u043e\u0440\u043e\u043c \u0432\u0441\u0435\u0445 \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445, \u043e\u0431\u043e\u0440\u0430\u0447\u0438\u0432\u0430\u044e\u0449\u0438\u0445 \u0433\u0440\u0430\u0444\u0438\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u041a\u0430\u0436\u0434\u0430\u044f \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u0430\u044f <img decoding=\"async\" class=\"formula inline\" source=\"y = kx + b\" alt=\"y = kx + b\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/532\/532b3d9f0bdbd776feaafd45c8a000d2.svg\" width=\"83\" height=\"16\" data-width=\"10.335\" data-height=\"2.034\" data-vertical-align=\"-0.464\" style=\"width: 10.335ex; height: 2.034ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/532\/532b3d9f0bdbd776feaafd45c8a000d2.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/532\/532b3d9f0bdbd776feaafd45c8a000d2.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0437\u0430\u0434\u0430\u0435\u0442\u0441\u044f \u0442\u043e\u0436\u0435 \u0434\u0432\u0443\u043c\u044f \u0447\u0438\u0441\u043b\u0430\u043c\u0438: <img decoding=\"async\" class=\"formula inline\" source=\"(k,b)\" alt=\"(k,b)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bdb\/bdbd61892c28f9f09762f00d41f9dfe4.svg\" width=\"39\" height=\"18\" data-width=\"4.916\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.916ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bdb\/bdbd61892c28f9f09762f00d41f9dfe4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bdb\/bdbd61892c28f9f09762f00d41f9dfe4.svg 781w\" loading=\"lazy\" decode=\"async\"\/> &#8212; \u043d\u0430\u043a\u043b\u043e\u043d\u043e\u043c \u0438 \u0441\u0434\u0432\u0438\u0433\u043e\u043c \u043f\u043e \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u0438<\/p>\n<\/li>\n<\/ul>\n<p>\u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0434\u043b\u044f \u0432\u044b\u043f\u0443\u043a\u043b\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u043e\u0431\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u044b. \u041f\u043e\u043a\u0430 \u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u043c, \u0430 \u0447\u0435\u043c \u044d\u0442\u043e \u043f\u043e\u043b\u0435\u0437\u043d\u043e &#8212; \u043d\u0438\u0436\u0435<\/p>\n<hr\/>\n<h2>\u0410\u043b\u0433\u0435\u0431\u0440\u0430<\/h2>\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u0443\u044e \u0445\u0438\u0442\u0440\u0443\u044e \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044e. \u0412\u043e\u0437\u044c\u043c\u0435\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"f(x)\" alt=\"f(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/50b\/50bbd36e1fd2333108437a2ca378be62.svg\" width=\"34\" height=\"18\" data-width=\"4.299\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.299ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/50b\/50bbd36e1fd2333108437a2ca378be62.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/50b\/50bbd36e1fd2333108437a2ca378be62.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u0441\u043e\u043f\u043e\u0441\u0442\u0430\u0432\u0438\u043c \u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p)\" alt=\"f^{*}(p)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg\" width=\"42\" height=\"18\" data-width=\"5.25\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.25ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e \u0442\u0430\u043a\u043e\u043c\u0443 \u043f\u0440\u0430\u0432\u0438\u043b\u0443 (\u043f\u043e\u043a\u0430 \u043f\u0440\u0438\u043c\u0435\u043c \u044d\u0442\u043e \u043a\u0430\u043a \u0447\u0435\u0440\u043d\u044b\u0439 \u044f\u0449\u0438\u043a, \u043f\u043e\u0441\u0442\u0435\u043f\u0435\u043d\u043d\u043e \u043f\u043e\u0439\u043c\u0435\u043c \u0433\u043b\u0443\u0431\u0438\u043d\u043d\u044b\u0435 \u0441\u043c\u044b\u0441\u043b\u044b):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f^{*}(p) = \\sup_{x}\\left( px - f(x) \\right)\" alt=\"f^{*}(p) = \\sup_{x}\\left( px - f(x) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/38\/387\/387ac3dfe58e9c4d4e692df7538cc3b8.svg\" width=\"186\" height=\"30\" data-width=\"23.308\" data-height=\"3.737\" data-vertical-align=\"-1.303\" style=\"width: 23.308ex; height: 3.737ex; vertical-align: -1.303ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/38\/387\/387ac3dfe58e9c4d4e692df7538cc3b8.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/38\/387\/387ac3dfe58e9c4d4e692df7538cc3b8.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0430\u0445:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) \\equiv 0\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}(px) = \\left\\{ \\begin{matrix} 0,\\ \\ p = 0 \\\\ \\infty,\\ \\ p \\neq 0 \\end{matrix} \\right.\" alt=\"f(x) \\equiv 0\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}(px) = \\left\\{ \\begin{matrix} 0,\\ \\ p = 0 \\\\ \\infty,\\ \\ p \\neq 0 \\end{matrix} \\right.\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9b\/9b0\/9b0dd451d8a82b425adf2295ccaa4a4a.svg\" width=\"356\" height=\"43\" data-width=\"44.495\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 44.495ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9b\/9b0\/9b0dd451d8a82b425adf2295ccaa4a4a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9b\/9b0\/9b0dd451d8a82b425adf2295ccaa4a4a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0438\u043b\u0430\u0441\u044c \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u0430\u044f \u00ab\u0432\u044b\u0432\u0435\u0440\u043d\u0443\u0442\u0430\u044f \u0434\u0435\u043b\u044c\u0442\u0430-\u0444\u0443\u043d\u043a\u0446\u0438\u044f\u00bb: \u0432\u043e \u0432\u0441\u0435\u0445 \u0442\u043e\u0447\u043a\u0430\u0445 <img decoding=\"async\" class=\"formula inline\" source=\"\\infty\" alt=\"\\infty\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7ed\/7ed9abff4dafd78d08e616c899412e92.svg\" width=\"18\" height=\"18\" data-width=\"2.262\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.262ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7ed\/7ed9abff4dafd78d08e616c899412e92.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7ed\/7ed9abff4dafd78d08e616c899412e92.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0432 \u043d\u0443\u043b\u0435 0. \u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u0441\u0434\u0432\u0438\u043d\u0443\u0442\u044c \u0438\u0441\u0445\u043e\u0434\u043d\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u044e f \u043f\u043e \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) \\equiv b\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}(px - b) = \\left\\{ \\begin{matrix} - b,\\ \\ p = 0 \\\\ \\infty,\\ \\ p \\neq 0 \\end{matrix} \\right.\" alt=\"f(x) \\equiv b\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}(px - b) = \\left\\{ \\begin{matrix} - b,\\ \\ p = 0 \\\\ \\infty,\\ \\ p \\neq 0 \\end{matrix} \\right.\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fee\/fee59cd839537b1dbe0c24ad311b22d3.svg\" width=\"388\" height=\"43\" data-width=\"48.539\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 48.539ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fee\/fee59cd839537b1dbe0c24ad311b22d3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fee\/fee59cd839537b1dbe0c24ad311b22d3.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u0430\u0448\u0430 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p)\" alt=\"f^{*}(p)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg\" width=\"42\" height=\"18\" data-width=\"5.25\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.25ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u0440\u043e\u0441\u0435\u043b\u0430 \u043d\u0430 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0441\u0434\u0432\u0438\u0433, \u043d\u043e \u0432\u0441\u0435 \u0435\u0449\u0435 \u0446\u0435\u043d\u0442\u0440\u0438\u0440\u043e\u0432\u0430\u043d\u0430 \u0432 <img decoding=\"async\" class=\"formula inline\" source=\"p = 0\" alt=\"p = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d0\/5d073ea1de9f64da4bb3f13e4bd15f80.svg\" width=\"42\" height=\"16\" data-width=\"5.286\" data-height=\"1.946\" data-vertical-align=\"-0.439\" style=\"width: 5.286ex; height: 1.946ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d0\/5d073ea1de9f64da4bb3f13e4bd15f80.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d0\/5d073ea1de9f64da4bb3f13e4bd15f80.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0423\u0441\u043b\u043e\u0436\u043d\u0438\u043c \u043f\u0440\u0438\u043c\u0435\u0440:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) = x\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( (p - 1)x \\right) = \\left\\{ \\begin{matrix} 0,\\ \\ p = 1 \\\\ \\infty,\\ \\ p \\neq 1 \\end{matrix} \\right.\" alt=\"f(x) = x\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( (p - 1)x \\right) = \\left\\{ \\begin{matrix} 0,\\ \\ p = 1 \\\\ \\infty,\\ \\ p \\neq 1 \\end{matrix} \\right.\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d2\/4d2f740a8a4b545aca0747f4943ff599.svg\" width=\"406\" height=\"43\" data-width=\"50.693\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 50.693ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d2\/4d2f740a8a4b545aca0747f4943ff599.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d2\/4d2f740a8a4b545aca0747f4943ff599.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0438\u043b\u0430\u0441\u044c \u0442\u0430 \u0436\u0435 \u00ab\u0432\u044b\u0432\u0435\u0440\u043d\u0443\u0442\u0430\u044f \u0434\u0435\u043b\u044c\u0442\u0430-\u0444\u0443\u043d\u043a\u0446\u0438\u044f\u00bb, \u043d\u043e \u0442\u0435\u043f\u0435\u0440\u044c \u0441 \u0446\u0435\u043d\u0442\u0440\u043e\u043c \u0432 1. \u0412\u0438\u0434\u043d\u043e, \u0447\u0442\u043e \u044d\u0442\u043e\u0442 \u043f\u0440\u0438\u043c\u0435\u0440 \u043e\u0442\u043b\u0438\u0447\u0430\u0435\u0442\u0441\u044f \u043e\u0442 \u0441\u0430\u043c\u043e\u0433\u043e \u043f\u0435\u0440\u0432\u043e\u0433\u043e \u0432 \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u043c \u043c\u0435\u0441\u0442\u0435: \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u043f\u0435\u0440\u0435\u0434 x \u0432 \u043f\u0435\u0440\u0432\u043e\u043c \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c\u0435 \u2013 \u043f\u0440\u043e\u0441\u0442\u043e p, \u0430 \u0432 \u0442\u0435\u043a\u0443\u0449\u0435\u043c \u2013 (p-1). \u042d\u0442\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0430 \u0432\u043e\u0437\u043d\u0438\u043a\u043b\u0430 \u0438\u043c\u0435\u043d\u043d\u043e \u043a\u0430\u043a \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u043f\u0435\u0440\u0435\u0434 x, \u0442\u043e \u0435\u0441\u0442\u044c \u043d\u0430\u043a\u043b\u043e\u043d \u0444\u0443\u043d\u043a\u0446\u0438\u0438 f. \u041e\u0431\u043e\u0431\u0449\u0438\u043c \u0438 \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u043c \u0438\u043d\u0442\u0443\u0438\u0446\u0438\u044e:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) = kx + b\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( (p - k)x - b \\right) = \\left\\{ \\begin{matrix} - b,\\ \\ p = k \\\\ \\infty,\\ \\ p \\neq k \\end{matrix} \\right.\" alt=\"f(x) = kx + b\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( (p - k)x - b \\right) = \\left\\{ \\begin{matrix} - b,\\ \\ p = k \\\\ \\infty,\\ \\ p \\neq k \\end{matrix} \\right.\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a5\/a53\/a53e3bd5a1c49dc6ac98843b8d0b02f9.svg\" width=\"479\" height=\"43\" data-width=\"59.907\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 59.907ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a5\/a53\/a53e3bd5a1c49dc6ac98843b8d0b02f9.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a5\/a53\/a53e3bd5a1c49dc6ac98843b8d0b02f9.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u0438\u0434\u043d\u043e, \u0447\u0442\u043e \u043d\u0430\u0448\u0430 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p)\" alt=\"f^{*}(p)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg\" width=\"42\" height=\"18\" data-width=\"5.25\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.25ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0434\u0435\u0442\u0435\u043a\u0442\u0438\u0440\u0443\u0435\u0442 \u043d\u0430\u043a\u043b\u043e\u043d \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 f(x). \u041a\u043e\u0433\u0434\u0430 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 p \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p)\" alt=\"f^{*}(p)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg\" width=\"42\" height=\"18\" data-width=\"5.25\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.25ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3da\/3da5c3fa39b421f43c4b5923625aaa05.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0435\u0442 \u0441 \u043d\u0430\u043a\u043b\u043e\u043d\u043e\u043c \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 f(x), \u0438\u043d\u0434\u0438\u043a\u0430\u0442\u043e\u0440 \u0441\u0440\u0430\u0431\u0430\u0442\u044b\u0432\u0430\u0435\u0442, \u0438 \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u043d\u0430 \u0432\u044b\u0445\u043e\u0434\u0435. \u0411\u043e\u043b\u0435\u0435 \u0442\u043e\u0433\u043e, \u044d\u0442\u043e \u0447\u0438\u0441\u043b\u043e <img decoding=\"async\" class=\"formula inline\" source=\"- b\" alt=\"- b\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/19\/191\/191e4643d492b35747e6f98f52aba65e.svg\" width=\"22\" height=\"14\" data-width=\"2.731\" data-height=\"1.756\" data-vertical-align=\"-0.186\" style=\"width: 2.731ex; height: 1.756ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/19\/191\/191e4643d492b35747e6f98f52aba65e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/19\/191\/191e4643d492b35747e6f98f52aba65e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441\u043e\u0432\u043f\u0430\u043b\u043e \u0441 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u043e\u0439, \u043d\u0430 \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043d\u0443\u0436\u043d\u043e \u0441\u0434\u0432\u0438\u043d\u0443\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"px\" alt=\"px\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/21d\/21de26caa6bcfc936378c4e45d235bd9.svg\" width=\"19\" height=\"12\" data-width=\"2.432\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 2.432ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/21d\/21de26caa6bcfc936378c4e45d235bd9.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/21d\/21de26caa6bcfc936378c4e45d235bd9.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u043d\u0438\u0437, \u0447\u0442\u043e\u0431\u044b \u00ab\u043e\u0431\u0435\u0440\u043d\u0443\u0442\u044c\u00bb \u0438\u0441\u0445\u043e\u0434\u043d\u0443\u044e f \u0441\u043d\u0438\u0437\u0443 (\u0438 \u0442\u043e\u0433\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\infty\" alt=\"\\infty\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7ed\/7ed9abff4dafd78d08e616c899412e92.svg\" width=\"18\" height=\"18\" data-width=\"2.262\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.262ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7ed\/7ed9abff4dafd78d08e616c899412e92.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7ed\/7ed9abff4dafd78d08e616c899412e92.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u0440\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"p \\neq k\" alt=\"p \\neq k\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0d\/0dc\/0dc3eb776bf83e4c0417806eac5272de.svg\" width=\"43\" height=\"18\" data-width=\"5.334\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.334ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0d\/0dc\/0dc3eb776bf83e4c0417806eac5272de.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0d\/0dc\/0dc3eb776bf83e4c0417806eac5272de.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043b\u043e\u0433\u0438\u0447\u043d\u0430, \u0432\u0435\u0434\u044c \u043f\u0440\u044f\u043c\u044b\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"y = px\" alt=\"y = px\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d2f\/d2f4bc2dac8484d205afd4eafdff6741.svg\" width=\"52\" height=\"14\" data-width=\"6.558\" data-height=\"1.783\" data-vertical-align=\"-0.464\" style=\"width: 6.558ex; height: 1.783ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d2f\/d2f4bc2dac8484d205afd4eafdff6741.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d2f\/d2f4bc2dac8484d205afd4eafdff6741.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"y = f(x)\" alt=\"y = f(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fd9\/fd91c508f91c2c84498680bd337c1d7a.svg\" width=\"67\" height=\"18\" data-width=\"8.424\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.424ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fd9\/fd91c508f91c2c84498680bd337c1d7a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fd9\/fd91c508f91c2c84498680bd337c1d7a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e \u043f\u0435\u0440\u0435\u0441\u0435\u043a\u0443\u0442\u0441\u044f). \u042d\u0442\u0430 \u0438\u0434\u0435\u044f \u0441\u043e\u0431\u0440\u0430\u043d\u0430 \u0432 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0435 \u0424\u0435\u043d\u0445\u0435\u043b\u044f \u2014 \u042e\u043d\u0433\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f^{}(p) = \\sup_{x}\\left( px - f(x) \\right)\\ \\\u00a0 \\Rightarrow \\ \\ f(x) \\geq px - f^{*}(p)\\ \\ \\forall x,p\" alt=\"f^{}(p) = \\sup_{x}\\left( px - f(x) \\right)\\ \\\u00a0 \\Rightarrow \\ \\ f(x) \\geq px - f^{*}(p)\\ \\ \\forall x,p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/be7\/be738bbb156e34a88dd624673c5d15cb.svg\" width=\"415\" height=\"30\" data-width=\"51.881\" data-height=\"3.737\" data-vertical-align=\"-1.303\" style=\"width: 51.881ex; height: 3.737ex; vertical-align: -1.303ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/be7\/be738bbb156e34a88dd624673c5d15cb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/be7\/be738bbb156e34a88dd624673c5d15cb.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u044f\u043c\u044b\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"px - f^{*}(p)\" alt=\"px - f^{*}(p)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8e\/8e3\/8e38932e49f7289f945945e3aa285a92.svg\" width=\"84\" height=\"18\" data-width=\"10.448\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 10.448ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8e\/8e3\/8e38932e49f7289f945945e3aa285a92.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8e\/8e3\/8e38932e49f7289f945945e3aa285a92.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0446\u0435\u043b\u0438\u043a\u043e\u043c \u043b\u0435\u0436\u0430\u0442 \u043f\u043e\u0434 \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u043c <img decoding=\"async\" class=\"formula inline\" source=\"f(x)\" alt=\"f(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/50b\/50bbd36e1fd2333108437a2ca378be62.svg\" width=\"34\" height=\"18\" data-width=\"4.299\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.299ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/50b\/50bbd36e1fd2333108437a2ca378be62.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/50b\/50bbd36e1fd2333108437a2ca378be62.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u043a\u0430\u043a \u0440\u0430\u0437 \u043d\u0430\u0431\u043e\u0440 \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445, \u043e\u0431\u043e\u0440\u0430\u0447\u0438\u0432\u0430\u044e\u0449\u0438\u0445 \u0437\u0430\u0434\u0430\u043d\u043d\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u044e f \u0441\u043d\u0438\u0437\u0443. \u041f\u0440\u043e\u0434\u043e\u043b\u0436\u0438\u043c \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u044b: \u0447\u0442\u043e \u0435\u0441\u043b\u0438 \u043d\u0430\u043a\u043b\u043e\u043d \u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \u0441\u043e \u0432\u0440\u0435\u043c\u0435\u043d\u0435\u043c?<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) = x^{2}\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( - x^{2} + px \\right)\" alt=\"f(x) = x^{2}\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( - x^{2} + px \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b85\/b8575ac9ad4d3364c9a29b3197fd3521.svg\" width=\"310\" height=\"32\" data-width=\"38.743\" data-height=\"4.04\" data-vertical-align=\"-1.454\" style=\"width: 38.743ex; height: 4.04ex; vertical-align: -1.454ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b85\/b8575ac9ad4d3364c9a29b3197fd3521.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b85\/b8575ac9ad4d3364c9a29b3197fd3521.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u043d\u0443\u0442\u0440\u0438 \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c\u0430 \u2013 \u043f\u0430\u0440\u0430\u0431\u043e\u043b\u0430 \u0432\u0435\u0442\u0432\u044f\u043c\u0438 \u0432\u043d\u0438\u0437. \u0422\u0435\u043f\u0435\u0440\u044c \u0441\u0432\u044f\u0437\u0430\u043b\u043e\u0441\u044c, \u043f\u043e\u0447\u0435\u043c\u0443 \u043c\u044b \u0432\u043e\u043e\u0431\u0449\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u043c \u0432\u044b\u043f\u0443\u043a\u043b\u044b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438: \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c \u0438\u0441\u043a\u0430\u0442\u044c \u043f\u0440\u043e\u0449\u0435. \u0412 \u0434\u0430\u043d\u043d\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c \u0434\u043e\u0441\u0442\u0438\u0433\u0430\u0435\u0442\u0441\u044f \u0432 \u0432\u0435\u0440\u0448\u0438\u043d\u0435 \u043f\u0430\u0440\u0430\u0431\u043e\u043b\u044b <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{p}{2}\" alt=\"\\frac{p}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/88b\/88b318400ecbd15bc8e689e712e73d48.svg\" width=\"17\" height=\"33\" data-width=\"2.133\" data-height=\"4.081\" data-vertical-align=\"-1.475\" style=\"width: 2.133ex; height: 4.081ex; vertical-align: -1.475ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/88b\/88b318400ecbd15bc8e689e712e73d48.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/88b\/88b318400ecbd15bc8e689e712e73d48.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"x = \\frac{p}{2}\" alt=\"x = \\frac{p}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4a\/f4a72026c55a5094de6311bfcfa938aa.svg\" width=\"52\" height=\"33\" data-width=\"6.445\" data-height=\"4.081\" data-vertical-align=\"-1.475\" style=\"width: 6.445ex; height: 4.081ex; vertical-align: -1.475ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4a\/f4a72026c55a5094de6311bfcfa938aa.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4a\/f4a72026c55a5094de6311bfcfa938aa.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442\u043e\u0433\u0434\u0430<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\ f^{*}(p) = px - f\\left( \\frac{p}{2} \\right) = p*\\frac{p}{2} - \\left( \\frac{p}{2} \\right)^{2} = \\frac{p^{2}}{4}\" alt=\"\\ f^{*}(p) = px - f\\left( \\frac{p}{2} \\right) = p*\\frac{p}{2} - \\left( \\frac{p}{2} \\right)^{2} = \\frac{p^{2}}{4}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e2\/e2e\/e2e92006ab2f100d54d551178b9e7973.svg\" width=\"349\" height=\"40\" data-width=\"43.639\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 43.639ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e2\/e2e\/e2e92006ab2f100d54d551178b9e7973.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e2\/e2e\/e2e92006ab2f100d54d551178b9e7973.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u0430\u0448\u0435\u0439 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0439 \u043f\u0430\u0440\u0430\u0431\u043e\u043b\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"f(x) = x^{2}\" alt=\"f(x) = x^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/929\/929ac65bb97a45bdbfa8cea3345dd790.svg\" width=\"77\" height=\"21\" data-width=\"9.598\" data-height=\"2.565\" data-vertical-align=\"-0.566\" style=\"width: 9.598ex; height: 2.565ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/929\/929ac65bb97a45bdbfa8cea3345dd790.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/929\/929ac65bb97a45bdbfa8cea3345dd790.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441\u043e\u043f\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043f\u0430\u0440\u0430\u0431\u043e\u043b\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p) = \\frac{p^{2}}{4}\" alt=\"f^{*}(p) = \\frac{p^{2}}{4}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d28\/d2807777663aa17b108a093141040471.svg\" width=\"91\" height=\"40\" data-width=\"11.388\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 11.388ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d28\/d2807777663aa17b108a093141040471.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d28\/d2807777663aa17b108a093141040471.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043b\u0435\u0436\u0430\u0449\u0430\u044f \u043d\u0438\u0436\u0435 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0439. \u0410 \u0447\u0442\u043e \u0435\u0441\u043b\u0438 \u043f\u0440\u043e\u0432\u0435\u0440\u043d\u0443\u0442\u044c \u043d\u0430\u0448\u0435 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0432 \u043e\u0431\u0440\u0430\u0442\u043d\u0443\u044e \u0441\u0442\u043e\u0440\u043e\u043d\u0443?<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) = \\frac{x^{2}}{4}\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( - \\frac{x^{2}}{4} + px \\right)\" alt=\"f(x) = \\frac{x^{2}}{4}\\ \\\u00a0 \\Rightarrow \\ \\ f^{*}(p) = \\sup_{x}\\left( - \\frac{x^{2}}{4} + px \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77c\/77c17c229890e6093078e9691cbd894c.svg\" width=\"336\" height=\"45\" data-width=\"41.992\" data-height=\"5.564\" data-vertical-align=\"-2.217\" style=\"width: 41.992ex; height: 5.564ex; vertical-align: -2.217ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77c\/77c17c229890e6093078e9691cbd894c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77c\/77c17c229890e6093078e9691cbd894c.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0410\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u044b\u0435 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u044f \u043f\u0440\u043e \u0432\u0435\u0440\u0448\u0438\u043d\u0443 \u043f\u0430\u0440\u0430\u0431\u043e\u043b\u044b \u043f\u0440\u0438\u0432\u0435\u0434\u0443\u0442 \u043d\u0430\u0441 \u043a \u0442\u043e\u043c\u0443, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p) = p^{2}\" alt=\"f^{*}(p) = p^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07e\/07e103d7632221c873b991230f8b8565.svg\" width=\"83\" height=\"21\" data-width=\"10.393\" data-height=\"2.565\" data-vertical-align=\"-0.566\" style=\"width: 10.393ex; height: 2.565ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07e\/07e103d7632221c873b991230f8b8565.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07e\/07e103d7632221c873b991230f8b8565.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f, \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u043e \u0432 \u044d\u0442\u043e\u043c \u043f\u0440\u0438\u043c\u0435\u0440\u0435 \u043c\u044b \u043f\u0440\u043e\u0432\u0435\u043b\u0438 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0434\u0432\u0430\u0436\u0434\u044b \u0438 \u0432\u0435\u0440\u043d\u0443\u043b\u0438\u0441\u044c \u043a \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"f^{**} = f\" alt=\"f^{**} = f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/884\/88457d03380d5761450df4849ab2613e.svg\" width=\"59\" height=\"17\" data-width=\"7.413\" data-height=\"2.142\" data-vertical-align=\"-0.464\" style=\"width: 7.413ex; height: 2.142ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/884\/88457d03380d5761450df4849ab2613e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/884\/88457d03380d5761450df4849ab2613e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0412\u0441\u0435\u0433\u0434\u0430 \u043b\u0438 \u044d\u0442\u043e \u0442\u0430\u043a? \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u043d\u0435\u0442: \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u0432\u044b\u043f\u0443\u043a\u043b\u044b\u0445 \u0438 \u0437\u0430\u043c\u043a\u043d\u0443\u0442\u044b\u0445 \u2013 \u044d\u0442\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0424\u0435\u043d\u0445\u0435\u043b\u044f-\u041c\u043e\u0440\u043e:<\/p>\n<ol>\n<li>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"f^{**} \\leq f\" alt=\"f^{**} \\leq f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/738\/7386c93f98ba19df079677a2bcf5be5d.svg\" width=\"59\" height=\"18\" data-width=\"7.413\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.413ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/738\/7386c93f98ba19df079677a2bcf5be5d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/738\/7386c93f98ba19df079677a2bcf5be5d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u0441\u0435\u0433\u0434\u0430. \u0424\u0435\u043d\u0445\u0435\u043b\u044c-\u042e\u043d\u0433: <img decoding=\"async\" class=\"formula inline\" source=\"f(x) \\geq px - f^{*}(p)\\ \\ \\forall x,p\\ \\\u00a0 \\Rightarrow \\ \\ f(x) \\geq \\sup_{p}{px - f^{*}(p)} = f^{**}(x)\" alt=\"f(x) \\geq px - f^{*}(p)\\ \\ \\forall x,p\\ \\\u00a0 \\Rightarrow \\ \\ f(x) \\geq \\sup_{p}{px - f^{*}(p)} = f^{**}(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b2\/b2d\/b2d1b2c14b98fa77aade695003fc26d4.svg\" width=\"481\" height=\"32\" data-width=\"60.144\" data-height=\"4.03\" data-vertical-align=\"-1.449\" style=\"width: 60.144ex; height: 4.03ex; vertical-align: -1.449ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b2\/b2d\/b2d1b2c14b98fa77aade695003fc26d4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b2\/b2d\/b2d1b2c14b98fa77aade695003fc26d4.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"f^{**}\" alt=\"f^{**}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7e9\/7e9c3ae1f49092a52d40c339a43bfc1d.svg\" width=\"25\" height=\"17\" data-width=\"3.152\" data-height=\"2.142\" data-vertical-align=\"-0.464\" style=\"width: 3.152ex; height: 2.142ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7e9\/7e9c3ae1f49092a52d40c339a43bfc1d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7e\/7e9\/7e9c3ae1f49092a52d40c339a43bfc1d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u0441\u0435\u0433\u0434\u0430 \u0432\u044b\u043f\u0443\u043a\u043b\u0430 \u0438 \u0437\u0430\u043c\u043a\u043d\u0443\u0442\u0430, \u043f\u043e \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044e<\/p>\n<\/li>\n<li>\n<p>\u0434\u043e\u0432\u043e\u043b\u044c\u043d\u043e \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e, \u043f\u043e \u043a\u043e\u0442\u043e\u0440\u043e\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"f^{**} = conv\\ f\" alt=\"f^{**} = conv\\ f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/868\/8687af4a623c5f7a5412c481f10f5fb4.svg\" width=\"100\" height=\"18\" data-width=\"12.511\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 12.511ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/868\/8687af4a623c5f7a5412c481f10f5fb4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/868\/8687af4a623c5f7a5412c481f10f5fb4.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f \u043e\u0431\u043e\u043b\u043e\u0447\u043a\u0430 f, \u0442.\u0435. \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0432\u0441\u0435\u0445 \u0445\u043e\u0440\u0434 \u0441 \u043a\u043e\u043d\u0446\u0430\u043c\u0438 \u0432 \u043d\u0430\u0434\u0433\u0440\u0430\u0444\u0438\u043a\u0435). \u041f\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u0447\u0435\u0440\u0435\u0437 \u0432\u044b\u043f\u0443\u043a\u043b\u044b\u0439 \u043d\u0430\u0434\u0433\u0440\u0430\u0444\u0438\u043a, \u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0435\u0442 \u0441\u043e \u0441\u0432\u043e\u0435\u0439 \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0439 \u043e\u0431\u043e\u043b\u043e\u0447\u043a\u043e\u0439, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0434\u043b\u044f \u0432\u044b\u043f\u0443\u043a\u043b\u044b\u0445 <img decoding=\"async\" class=\"formula inline\" source=\"f^{**} = f\" alt=\"f^{**} = f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/884\/88457d03380d5761450df4849ab2613e.svg\" width=\"59\" height=\"17\" data-width=\"7.413\" data-height=\"2.142\" data-vertical-align=\"-0.464\" style=\"width: 7.413ex; height: 2.142ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/884\/88457d03380d5761450df4849ab2613e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/884\/88457d03380d5761450df4849ab2613e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<\/ol>\n<p>\u0417\u0430\u043c\u0435\u0442\u0438\u043c \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e, \u0447\u0442\u043e, \u043e\u043f\u0438\u0440\u0430\u044f\u0441\u044c \u043d\u0430 \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u044c, \u043c\u044b \u043f\u043e\u0434\u0431\u0438\u0440\u0430\u043b\u0438 \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c \u043f\u043e \u043d\u0443\u043b\u044f\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u043e\u0439: <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p) = \\left( px - f(x) \\right)\\left. \\\u00a0 \\right|_{p = f^{'}(x)} = \\left( px - f(x) \\right)\\left. \\\u00a0 \\right|_{x = \\left( f^{'} \\right)^{- 1}(p)}\" alt=\"f^{*}(p) = \\left( px - f(x) \\right)\\left. \\\u00a0 \\right|_{p = f^{'}(x)} = \\left( px - f(x) \\right)\\left. \\\u00a0 \\right|_{x = \\left( f^{'} \\right)^{- 1}(p)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7a\/7a4\/7a4c3a21c084a941e4d68b933746f52d.svg\" width=\"408\" height=\"26\" data-width=\"51.029\" data-height=\"3.195\" data-vertical-align=\"-1.032\" style=\"width: 51.029ex; height: 3.195ex; vertical-align: -1.032ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7a\/7a4\/7a4c3a21c084a941e4d68b933746f52d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7a\/7a4\/7a4c3a21c084a941e4d68b933746f52d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><em>.<\/em> \u0422\u0430\u043a\u0430\u044f \u0437\u0430\u043c\u0435\u043d\u0430 \u0442\u0440\u0435\u0431\u0443\u0435\u0442 \u043e\u0431\u0440\u0430\u0442\u0438\u043c\u043e\u0441\u0442\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"f^{'}\" alt=\"f^{'}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/25a\/25ac8ac9a5511fec40dea8bb48f21113.svg\" width=\"16\" height=\"21\" data-width=\"1.996\" data-height=\"2.612\" data-vertical-align=\"-0.464\" style=\"width: 1.996ex; height: 2.612ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/25a\/25ac8ac9a5511fec40dea8bb48f21113.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/25a\/25ac8ac9a5511fec40dea8bb48f21113.svg 781w\" loading=\"lazy\" decode=\"async\"\/><em>,<\/em> \u0442.\u0435. \u0441\u0442\u0440\u043e\u0433\u043e\u0439 \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u0438 \u2013 \u044d\u0442\u043e \u043d\u0430\u043c \u043f\u043e\u0437\u0436\u0435 \u043f\u0440\u0438\u0433\u043e\u0434\u0438\u0442\u0441\u044f.<\/p>\n<p>\u041f\u043e\u0434\u0443\u043c\u0430\u0435\u043c, \u0447\u0442\u043e \u0438\u043c\u0435\u043d\u043d\u043e \u0434\u0435\u043b\u0430\u0435\u0442 \u043d\u0430\u0448\u0435 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435. \u041f\u043e \u0433\u0440\u0430\u0444\u0438\u043a\u0430\u043c \u0440\u0430\u043d\u0435\u0435 \u0431\u044b\u043b\u043e \u0432\u0438\u0434\u043d\u043e, \u0447\u0442\u043e \u043a\u0440\u0438\u0432\u044b\u0435 \u043a\u0430\u043a \u0431\u0443\u0434\u0442\u043e \u043e\u0442\u0440\u0430\u0436\u0430\u044e\u0442\u0441\u044f \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u0433\u0440\u0430\u043d\u0438\u0446\u044b \u043c\u0435\u0436\u0434\u0443 \u043d\u0438\u043c\u0438 \u2013 \u0447\u0442\u043e \u044d\u0442\u043e \u0437\u0430 \u0433\u0440\u0430\u043d\u0438\u0446\u0430? \u0423\u0433\u0430\u0434\u0430\u0435\u043c \u043d\u0435\u043f\u043e\u0434\u0432\u0438\u0436\u043d\u0443\u044e \u0442\u043e\u0447\u043a\u0443:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) = \\frac{x^{2}}{2},\\ \\ f^{*}(p) = \\sup_{x}\\left( - \\frac{x^{2}}{2} + px \\right) = \\left( - \\frac{x^{2}}{2} + px \\right)\\left. \\\u00a0 \\right|_{p = f^{'}(x)} = \\frac{p^{2}}{2}\" alt=\"f(x) = \\frac{x^{2}}{2},\\ \\ f^{*}(p) = \\sup_{x}\\left( - \\frac{x^{2}}{2} + px \\right) = \\left( - \\frac{x^{2}}{2} + px \\right)\\left. \\\u00a0 \\right|_{p = f^{'}(x)} = \\frac{p^{2}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6f\/6f3\/6f3d9a8ccff788b20bbe4f7a1a9e1ce1.svg\" width=\"545\" height=\"45\" data-width=\"68.162\" data-height=\"5.564\" data-vertical-align=\"-2.217\" style=\"width: 68.162ex; height: 5.564ex; vertical-align: -2.217ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6f\/6f3\/6f3d9a8ccff788b20bbe4f7a1a9e1ce1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6f\/6f3\/6f3d9a8ccff788b20bbe4f7a1a9e1ce1.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0434\u0440\u0443\u0433\u0430\u044f \u043d\u0435\u043f\u043e\u0434\u0432\u0438\u0436\u043d\u0430\u044f \u0442\u043e\u0447\u043a\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"g = g^{*},\\ \\ g \\neq f\" alt=\"g = g^{*},\\ \\ g \\neq f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f57\/f57a17b391e891718cd95de6d8ab0aac.svg\" width=\"109\" height=\"18\" data-width=\"13.641\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.641ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f57\/f57a17b391e891718cd95de6d8ab0aac.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f57\/f57a17b391e891718cd95de6d8ab0aac.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0424\u0435\u043d\u0445\u0435\u043b\u044c-\u042e\u043d\u0433: <img decoding=\"async\" class=\"formula inline\" source=\"px \\leq g(x) + g^{*}(p) = g(x) + g(p)\" alt=\"px \\leq g(x) + g^{*}(p) = g(x) + g(p)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/101\/101e77e50ecf4abb024d537b237ab300.svg\" width=\"250\" height=\"18\" data-width=\"31.207\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 31.207ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/101\/101e77e50ecf4abb024d537b237ab300.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/101\/101e77e50ecf4abb024d537b237ab300.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0434\u043b\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\forall x,p\" alt=\"\\forall x,p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/12\/124\/124f79c475bafb8b212372f774b17229.svg\" width=\"38\" height=\"18\" data-width=\"4.696\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.696ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/12\/124\/124f79c475bafb8b212372f774b17229.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/12\/124\/124f79c475bafb8b212372f774b17229.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412 \u0447\u0430\u0441\u0442\u043d\u043e\u0441\u0442\u0438, \u0434\u043b\u044f <img decoding=\"async\" class=\"formula inline\" source=\"x = p\" alt=\"x = p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f6a\/f6ab8010df4a12328221b63415ac46cb.svg\" width=\"44\" height=\"14\" data-width=\"5.449\" data-height=\"1.758\" data-vertical-align=\"-0.439\" style=\"width: 5.449ex; height: 1.758ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f6a\/f6ab8010df4a12328221b63415ac46cb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f6a\/f6ab8010df4a12328221b63415ac46cb.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e\u043b\u0443\u0447\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{x^{2}}{2} &lt; g(x)\" alt=\"\\frac{x^{2}}{2} &lt; g(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b9\/b96\/b96dadf7d38c87a71ccec4fb3c0aa8a2.svg\" width=\"83\" height=\"40\" data-width=\"10.428\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 10.428ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b9\/b96\/b96dadf7d38c87a71ccec4fb3c0aa8a2.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b9\/b96\/b96dadf7d38c87a71ccec4fb3c0aa8a2.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041d\u043e \u0442\u043e\u0433\u0434\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"g^{*}(p) = \\sup_{x}\\left( px - g(x) \\right) &lt; \\sup_{x}\\left( px - \\frac{x^{2}}{2} \\right) = \\frac{p^{2}}{2}\" alt=\"g^{*}(p) = \\sup_{x}\\left( px - g(x) \\right) &lt; \\sup_{x}\\left( px - \\frac{x^{2}}{2} \\right) = \\frac{p^{2}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/79\/79a\/79abf92bb84451d986b20b77c7e336a6.svg\" width=\"381\" height=\"45\" data-width=\"47.603\" data-height=\"5.564\" data-vertical-align=\"-2.217\" style=\"width: 47.603ex; height: 5.564ex; vertical-align: -2.217ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/79\/79a\/79abf92bb84451d986b20b77c7e336a6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/79\/79a\/79abf92bb84451d986b20b77c7e336a6.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"g \\neq g^{*}\" alt=\"g \\neq g^{*}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/503\/5030c83204b3d8eb85310b89995e9d7f.svg\" width=\"49\" height=\"18\" data-width=\"6.163\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.163ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/503\/5030c83204b3d8eb85310b89995e9d7f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/50\/503\/5030c83204b3d8eb85310b89995e9d7f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u0440\u0435\u0447\u0438\u0435. \u0417\u043d\u0430\u0447\u0438\u0442, <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{x^{2}}{2}\" alt=\"\\frac{x^{2}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea758de61770df07aeeaac5ce2915664.svg\" width=\"26\" height=\"40\" data-width=\"3.277\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 3.277ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea758de61770df07aeeaac5ce2915664.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea758de61770df07aeeaac5ce2915664.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u043d\u0435\u043f\u043e\u0434\u0432\u0438\u0436\u043d\u0430\u044f \u0442\u043e\u0447\u043a\u0430. \u041f\u043e \u0441\u0443\u0442\u0438, \u043f\u0440\u043e\u0432\u043e\u0434\u044f \u043d\u0430\u0448\u0435 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435, \u043c\u044b \u00ab\u043e\u0442\u0440\u0430\u0436\u0430\u043b\u0438\u00bb \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u00ab\u0437\u0435\u0440\u043a\u0430\u043b\u0430\u00bb <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{x^{2}}{2}\" alt=\"\\frac{x^{2}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea758de61770df07aeeaac5ce2915664.svg\" width=\"26\" height=\"40\" data-width=\"3.277\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 3.277ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea758de61770df07aeeaac5ce2915664.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea758de61770df07aeeaac5ce2915664.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u041a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043c\u044b \u0432\u0441\u0435 \u044d\u0442\u043e \u0432\u0440\u0435\u043c\u044f \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u043b\u0438, \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430. \u041a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u0438\u0434\u0435\u044f: \u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f = \u0432\u0435\u0440\u0445\u043d\u044f\u044f \u043e\u0433\u0438\u0431\u0430\u044e\u0449\u0430\u044f \u0441\u0432\u043e\u0438\u0445 \u043d\u0438\u0436\u043d\u0438\u0445 \u043e\u0446\u0435\u043d\u043e\u043a.<\/p>\n<hr\/>\n<h2>\u0424\u0438\u0437\u0438\u043a\u0430 \/ \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0430<\/h2>\n<p>\u0421\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0442\u0440\u0438 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u043f\u043e\u0434\u0445\u043e\u0434\u0430 \u0434\u043b\u044f \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0441\u0438\u0441\u0442\u0435\u043c: \u043f\u043e \u041d\u044c\u044e\u0442\u043e\u043d\u0443, \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0443 \u0438 \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0443. \u041d\u044c\u044e\u0442\u043e\u043d \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0441 \u0441\u0438\u043b\u0430\u043c\u0438 \u2013 \u043d\u0430\u043c \u044d\u0442\u043e \u0441\u0435\u0439\u0447\u0430\u0441 \u043d\u0435 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u043e. \u0421\u0444\u043e\u043a\u0443\u0441\u0438\u0440\u0443\u0435\u043c\u0441\u044f \u043d\u0430 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0435\u0432\u043e\u043c \u0438 \u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u043e\u0432\u043e\u043c \u0444\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u043c\u0435, \u043f\u043e\u043a\u0430\u0436\u0435\u043c \u0441\u0432\u044f\u0437\u044c \u043c\u0435\u0436\u0434\u0443 \u043d\u0438\u043c\u0438 \u0438 \u043c\u0435\u0441\u0442\u043e, \u0433\u0434\u0435 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430.<\/p>\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0437\u0430\u0434\u0430\u0447\u0443 \u043a\u0438\u043d\u0435\u043c\u0430\u0442\u0438\u043a\u0438: \u043f\u043e \u043a\u0430\u043a\u043e\u0439 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438 \u043f\u043e\u043b\u0435\u0442\u0438\u0442 \u043a\u0430\u043c\u0435\u043d\u044c \u0432 \u043f\u043e\u043b\u0435 \u0433\u0440\u0430\u0432\u0438\u0442\u0430\u0446\u0438\u0438 (\u0434\u043b\u044f \u043f\u0440\u043e\u0441\u0442\u043e\u0442\u044b \u2013 \u0431\u0440\u043e\u0441\u0438\u043c \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u044c\u043d\u043e)?<\/p>\n<p>\u041c\u043e\u0436\u043d\u043e \u0432\u0432\u0435\u0441\u0442\u0438 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"q\" alt=\"q\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg\" width=\"12\" height=\"12\" data-width=\"1.041\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.041ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u0443\u044e \u044d\u043d\u0435\u0440\u0433\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"U = mgq\" alt=\"U = mgq\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/72\/72a\/72ae13c3e3f4a5a6a9b499c52b7b055e.svg\" width=\"71\" height=\"16\" data-width=\"8.859\" data-height=\"2.009\" data-vertical-align=\"-0.464\" style=\"width: 8.859ex; height: 2.009ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/72\/72a\/72ae13c3e3f4a5a6a9b499c52b7b055e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/72\/72a\/72ae13c3e3f4a5a6a9b499c52b7b055e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043a\u0438\u043d\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0443\u044e \u044d\u043d\u0435\u0440\u0433\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"T = \\frac{mv^{2}}{2} = \\frac{m{\\dot{q}}^{2}}{2}\" alt=\"T = \\frac{mv^{2}}{2} = \\frac{m{\\dot{q}}^{2}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c35\/c35327133cdb7ddeabe09ea3528c71bf.svg\" width=\"143\" height=\"40\" data-width=\"17.819\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 17.819ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c35\/c35327133cdb7ddeabe09ea3528c71bf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c35\/c35327133cdb7ddeabe09ea3528c71bf.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412 \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435 \u0432\u044b\u043f\u0443\u043a\u043b\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u0441\u0440\u0430\u0437\u0443 \u0431\u0440\u043e\u0441\u0430\u0435\u0442\u0441\u044f \u0432 \u0433\u043b\u0430\u0437\u0430 \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u044c \u043a\u0438\u043d\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u044d\u043d\u0435\u0440\u0433\u0438\u0438 \u043f\u043e \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044f\u043c (\u0441\u043f\u043e\u0439\u043b\u0435\u0440: \u044d\u0442\u043e \u043f\u0440\u0438\u0433\u043e\u0434\u0438\u0442\u0441\u044f).<\/p>\n<p>\u041f\u043e\u043b\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u0441\u043e\u0445\u0440\u0430\u043d\u044f\u0435\u0442\u0441\u044f, \u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"0 = \\frac{d}{dt}\\left( mgq + \\frac{m{\\dot{q}}^{2}}{2} \\right) = mg\\dot{q} + m\\dot{q}\\ddot{q} = m\\dot{q}(g + \\ddot{q})\" alt=\"0 = \\frac{d}{dt}\\left( mgq + \\frac{m{\\dot{q}}^{2}}{2} \\right) = mg\\dot{q} + m\\dot{q}\\ddot{q} = m\\dot{q}(g + \\ddot{q})\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/826\/826e7e79410a3fb087680622aaac2a89.svg\" width=\"393\" height=\"45\" data-width=\"49.111\" data-height=\"5.564\" data-vertical-align=\"-2.217\" style=\"width: 49.111ex; height: 5.564ex; vertical-align: -2.217ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/826\/826e7e79410a3fb087680622aaac2a89.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/826\/826e7e79410a3fb087680622aaac2a89.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u0440\u0438\u0447\u0435\u043c \u043c\u0430\u0441\u0441\u0430 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u0430, \u0430 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0437\u0430\u043d\u0443\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u043e\u043b\u044c\u043a\u043e \u0432 \u043e\u0434\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u0435 (\u0432\u0435\u0440\u0448\u0438\u043d\u0435 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438). \u041e\u0441\u0442\u0430\u0435\u0442\u0441\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\ddot{q} = - g\" alt=\"\\ddot{q} = - g\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1c\/1c7\/1c7914470f52aa2970be53c7fa9e9f60.svg\" width=\"55\" height=\"18\" data-width=\"6.897\" data-height=\"2.29\" data-vertical-align=\"-0.579\" style=\"width: 6.897ex; height: 2.29ex; vertical-align: -0.579ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1c\/1c7\/1c7914470f52aa2970be53c7fa9e9f60.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1c\/1c7\/1c7914470f52aa2970be53c7fa9e9f60.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0420\u0435\u0448\u0438\u0432 \u044d\u0442\u043e\u0442 \u0434\u0438\u0444\u0443\u0440, \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u0437\u043d\u0430\u043a\u043e\u043c\u043e\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"q = q_{0} + v_{0}t - \\frac{gt^{2}}{2}\" alt=\"q = q_{0} + v_{0}t - \\frac{gt^{2}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a8\/a89\/a89209b0626ecb3f0e6d0eb89a7b16c5.svg\" width=\"147\" height=\"40\" data-width=\"18.367\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 18.367ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a8\/a89\/a89209b0626ecb3f0e6d0eb89a7b16c5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a8\/a89\/a89209b0626ecb3f0e6d0eb89a7b16c5.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0412\u043c\u0435\u0441\u0442\u043e \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q}\" alt=\"\\dot{q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg\" width=\"12\" height=\"18\" data-width=\"1.041\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 1.041ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043c\u043e\u0436\u043d\u043e \u043e\u043f\u0438\u0441\u0430\u0442\u044c \u0441\u0438\u0441\u0442\u0435\u043c\u0443 \u0447\u0435\u0440\u0435\u0437 \u0438\u0437\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u0430: <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{dp}{dt} = - mg,\\ \\ p = m\\dot{q} = mv_{0} - mgt\" alt=\"\\frac{dp}{dt} = - mg,\\ \\ p = m\\dot{q} = mv_{0} - mgt\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4bd\/4bd2583d394d931da5e1eb6ebdf77f06.svg\" width=\"274\" height=\"37\" data-width=\"34.209\" data-height=\"4.676\" data-vertical-align=\"-1.773\" style=\"width: 34.209ex; height: 4.676ex; vertical-align: -1.773ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4bd\/4bd2583d394d931da5e1eb6ebdf77f06.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4bd\/4bd2583d394d931da5e1eb6ebdf77f06.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0421\u043e\u043a\u0440\u0430\u0442\u0438\u0432, \u043f\u043e\u043b\u0443\u0447\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q} = v_{0} - gt\" alt=\"\\dot{q} = v_{0} - gt\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a492cdd9addc7fc77a72b68ef5aa4526.svg\" width=\"86\" height=\"18\" data-width=\"10.804\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 10.804ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a492cdd9addc7fc77a72b68ef5aa4526.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a492cdd9addc7fc77a72b68ef5aa4526.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0420\u0435\u0448\u0438\u0432 \u044d\u0442\u043e\u0442 \u0434\u0438\u0444\u0443\u0440, \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u0442\u0430\u043a\u043e\u0439 \u0436\u0435 \u043e\u0442\u0432\u0435\u0442: <img decoding=\"async\" class=\"formula inline\" source=\"q = q_{0} + v_{0}t - \\frac{gt^{2}}{2}\" alt=\"q = q_{0} + v_{0}t - \\frac{gt^{2}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a8\/a89\/a89209b0626ecb3f0e6d0eb89a7b16c5.svg\" width=\"147\" height=\"40\" data-width=\"18.367\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 18.367ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a8\/a89\/a89209b0626ecb3f0e6d0eb89a7b16c5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a8\/a89\/a89209b0626ecb3f0e6d0eb89a7b16c5.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0412\u044b\u0432\u043e\u0434: \u0434\u0432\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0441\u0438\u0441\u0442\u0435\u043c\u044b (\u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0438 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0438 vs \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0438 \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u044b) \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u044b \u0432 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043d\u043d\u043e\u0439 \u043d\u0430\u043c\u0438 \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0435.<\/p>\n<p>\u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u043e\u0431\u0430 \u043f\u043e\u0434\u0445\u043e\u0434\u0430 \u043e\u043f\u0438\u0440\u0430\u044e\u0442\u0441\u044f \u043d\u0430 \u0431\u043e\u043b\u0435\u0435 \u0433\u043b\u0443\u0431\u043e\u043a\u0443\u044e \u0442\u0435\u043e\u0440\u0438\u044e \u0438 \u0442\u0435\u0441\u043d\u043e \u0441\u0432\u044f\u0437\u0430\u043d\u044b<\/p>\n<p>\u041b\u0430\u0433\u0440\u0430\u043d\u0436 \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442 \u0441\u0438\u0441\u0442\u0435\u043c\u0443 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u043d\u044b\u0445 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"q\" alt=\"q\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg\" width=\"12\" height=\"12\" data-width=\"1.041\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.041ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0435\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q}\" alt=\"\\dot{q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg\" width=\"12\" height=\"18\" data-width=\"1.041\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 1.041ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u043a\u0438\u043d\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"T\\left( q,\\dot{q} \\right)\" alt=\"T\\left( q,\\dot{q} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/69d\/69d21a9db44a092a70efb5cad6df6985.svg\" width=\"55\" height=\"18\" data-width=\"6.817\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 6.817ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/69d\/69d21a9db44a092a70efb5cad6df6985.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/69d\/69d21a9db44a092a70efb5cad6df6985.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"U\\left( q,\\dot{q} \\right)\" alt=\"U\\left( q,\\dot{q} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8e\/8e3\/8e35deec3c69df53c9885be35dbc870d.svg\" width=\"56\" height=\"18\" data-width=\"6.96\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 6.96ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8e\/8e3\/8e35deec3c69df53c9885be35dbc870d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8e\/8e3\/8e35deec3c69df53c9885be35dbc870d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0438\u0445 \u0440\u0430\u0437\u043d\u043e\u0441\u0442\u044c \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d\u043e\u043c <img decoding=\"async\" class=\"formula inline\" source=\"L = T - U\" alt=\"L = T - U\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba5\/ba50da65ec179d38569df6a44747e5e3.svg\" width=\"85\" height=\"14\" data-width=\"10.652\" data-height=\"1.731\" data-vertical-align=\"-0.186\" style=\"width: 10.652ex; height: 1.731ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba5\/ba50da65ec179d38569df6a44747e5e3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba5\/ba50da65ec179d38569df6a44747e5e3.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0418\u043d\u0442\u0435\u0433\u0440\u0430\u043b \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d\u0430 \u043f\u043e \u0432\u0440\u0435\u043c\u0435\u043d\u0438 (\u0432\u0434\u043e\u043b\u044c \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438) \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0435\u043c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"S\\lbrack q\\rbrack = \\int_{t_{1}}^{t_{2}}{L\\left( q,\\dot{q},t \\right)dt}\" alt=\"S\\lbrack q\\rbrack = \\int_{t_{1}}^{t_{2}}{L\\left( q,\\dot{q},t \\right)dt}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/455\/45548feb3743d0bffac6080497e5767a.svg\" width=\"172\" height=\"46\" data-width=\"21.49\" data-height=\"5.731\" data-vertical-align=\"-2.3\" style=\"width: 21.49ex; height: 5.731ex; vertical-align: -2.3ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/455\/45548feb3743d0bffac6080497e5767a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/455\/45548feb3743d0bffac6080497e5767a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b, \u0441\u043e\u043f\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0449\u0438\u0439 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"q(t)\" alt=\"q(t)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c9e\/c9ef9cf8733ebf3f233aa6eafda45d88.svg\" width=\"29\" height=\"18\" data-width=\"3.618\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.618ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c9e\/c9ef9cf8733ebf3f233aa6eafda45d88.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c9e\/c9ef9cf8733ebf3f233aa6eafda45d88.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u041f\u043e\u0441\u0442\u0443\u043b\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u041f\u0440\u0438\u043d\u0446\u0438\u043f \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0435\u0433\u043e \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f: \u043f\u0440\u0438\u0440\u043e\u0434\u0430 \u0441\u0442\u0440\u0435\u043c\u0438\u0442\u0441\u044f \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u044d\u0442\u043e\u0442 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438 \u0442\u043e\u0447\u0435\u043a \u2013 \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u0438. \u041f\u043e\u0447\u0435\u043c\u0443 \u044d\u0442\u043e \u0432\u0435\u0440\u043d\u043e \u2013 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u0439 \u0433\u043b\u0443\u0431\u043e\u043a\u0438\u0439 \u0432\u043e\u043f\u0440\u043e\u0441.<\/p>\n<figure class=\"full-width \"><img decoding=\"async\" src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/744\/b9c\/1cc\/744b9c1cc0e252381da5404d73011263.png\" width=\"556\" height=\"357\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/744\/b9c\/1cc\/744b9c1cc0e252381da5404d73011263.png 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/744\/b9c\/1cc\/744b9c1cc0e252381da5404d73011263.png 781w\" loading=\"lazy\" decode=\"async\"\/><\/figure>\n<p>\u0412\u043e\u0437\u043c\u0443\u0442\u0438\u043c \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u044e (\u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043d\u0438\u044f \u043a \u0432\u043e\u0437\u043c\u0443\u0449\u0435\u043d\u0438\u044e \u2013 \u0433\u043b\u0430\u0434\u043a\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f, \u043e\u0431\u043d\u0443\u043b\u044f\u044e\u0449\u0430\u044f\u0441\u044f \u043d\u0430 \u043a\u043e\u043d\u0446\u0430\u0445):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"q_{\\varepsilon}(t) = q(t) + \\varepsilon \\cdot \\eta(t),\\ \\ {\\dot{q}}_{\\varepsilon}(t) = \\dot{q}(t) + \\varepsilon \\cdot \\dot{\\eta}(t),\\ \\ \\eta\\left( t_{1} \\right) = \\eta\\left( t_{2} \\right) = 0\" alt=\"q_{\\varepsilon}(t) = q(t) + \\varepsilon \\cdot \\eta(t),\\ \\ {\\dot{q}}_{\\varepsilon}(t) = \\dot{q}(t) + \\varepsilon \\cdot \\dot{\\eta}(t),\\ \\ \\eta\\left( t_{1} \\right) = \\eta\\left( t_{2} \\right) = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/def\/def0fc653e1653e24447a544ba2c6fe5.svg\" width=\"498\" height=\"18\" data-width=\"62.192\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 62.192ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/def\/def0fc653e1653e24447a544ba2c6fe5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/def\/def0fc653e1653e24447a544ba2c6fe5.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0414\u0435\u0439\u0441\u0442\u0432\u0438\u0435 \u0432\u0434\u043e\u043b\u044c \u0432\u043e\u0437\u043c\u0443\u0449\u0435\u043d\u043d\u043e\u0439 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"S\\left\\lbrack q_{\\varepsilon} \\right\\rbrack = S(\\varepsilon) = \\int_{t_{1}}^{t_{2}}{L\\left( q + \\varepsilon\\eta,\\dot{q} + \\varepsilon\\dot{\\eta},t \\right)dt}\" alt=\"S\\left\\lbrack q_{\\varepsilon} \\right\\rbrack = S(\\varepsilon) = \\int_{t_{1}}^{t_{2}}{L\\left( q + \\varepsilon\\eta,\\dot{q} + \\varepsilon\\dot{\\eta},t \\right)dt}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/de1\/de191fada84091f5837aea9f102f99db.svg\" width=\"320\" height=\"46\" data-width=\"39.948\" data-height=\"5.731\" data-vertical-align=\"-2.3\" style=\"width: 39.948ex; height: 5.731ex; vertical-align: -2.3ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/de1\/de191fada84091f5837aea9f102f99db.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/de1\/de191fada84091f5837aea9f102f99db.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u043e\u0432\u0430\u0440\u044c\u0438\u0440\u0443\u0435\u043c: \u043f\u0440\u043e\u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c \u043f\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\varepsilon\" alt=\"\\varepsilon\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8b\/f8b1c5a729a09649c275fca88976d8dd.svg\" width=\"12\" height=\"18\" data-width=\"1.054\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.054ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8b\/f8b1c5a729a09649c275fca88976d8dd.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8b\/f8b1c5a729a09649c275fca88976d8dd.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043d\u0430\u0439\u0434\u0435\u043c \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"0 = \\frac{\\delta S}{\\delta q} = \\frac{dS}{d\\varepsilon} = \\int_{t_{1}}^{t_{2}}{\\frac{dL\\left( q + \\varepsilon\\eta,\\dot{q} + \\varepsilon\\dot{\\eta},t \\right)}{d\\varepsilon}dt} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q}\\frac{\\partial q_{\\varepsilon}}{\\partial\\varepsilon} + \\frac{\\partial L}{\\partial\\dot{q}}\\frac{\\partial{\\dot{q}}_{\\varepsilon}}{\\partial\\varepsilon} \\right\\rbrack dt} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q}\\eta \\right\\rbrack dt} + \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial\\dot{q}}\\dot{\\eta} \\right\\rbrack dt}\" alt=\"0 = \\frac{\\delta S}{\\delta q} = \\frac{dS}{d\\varepsilon} = \\int_{t_{1}}^{t_{2}}{\\frac{dL\\left( q + \\varepsilon\\eta,\\dot{q} + \\varepsilon\\dot{\\eta},t \\right)}{d\\varepsilon}dt} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q}\\frac{\\partial q_{\\varepsilon}}{\\partial\\varepsilon} + \\frac{\\partial L}{\\partial\\dot{q}}\\frac{\\partial{\\dot{q}}_{\\varepsilon}}{\\partial\\varepsilon} \\right\\rbrack dt} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q}\\eta \\right\\rbrack dt} + \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial\\dot{q}}\\dot{\\eta} \\right\\rbrack dt}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a25\/a2537cf6c669c188da2f4ab9b3a0fb0a.svg\" width=\"857\" height=\"46\" data-width=\"107.115\" data-height=\"5.731\" data-vertical-align=\"-2.3\" style=\"width: 107.115ex; height: 5.731ex; vertical-align: -2.3ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a25\/a2537cf6c669c188da2f4ab9b3a0fb0a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a25\/a2537cf6c669c188da2f4ab9b3a0fb0a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u0442\u043e\u0440\u043e\u0435 \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u043e\u0435 \u2013 \u043f\u0440\u043e\u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0443\u0435\u043c \u043f\u043e \u0447\u0430\u0441\u0442\u044f\u043c, \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u0432, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\eta\\left( t_{1} \\right) = \\eta\\left( t_{2} \\right) = 0\" alt=\"\\eta\\left( t_{1} \\right) = \\eta\\left( t_{2} \\right) = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/83a\/83a18593bf1c6cdaeacf3b3329a61a4d.svg\" width=\"138\" height=\"18\" data-width=\"17.298\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 17.298ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/83a\/83a18593bf1c6cdaeacf3b3329a61a4d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/83a\/83a18593bf1c6cdaeacf3b3329a61a4d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"0 = \\frac{dS}{d\\varepsilon} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q}\\eta \\right\\rbrack dt} + \\left\\lbrack \\frac{\\partial L}{\\partial\\dot{q}}\\eta \\right\\rbrack_{t_{1}}^{t_{2}} - \\int_{t_{1}}^{t_{2}}{\\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right)\\eta dt} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q} - \\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right) \\right\\rbrack\\eta dt}\" alt=\"0 = \\frac{dS}{d\\varepsilon} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q}\\eta \\right\\rbrack dt} + \\left\\lbrack \\frac{\\partial L}{\\partial\\dot{q}}\\eta \\right\\rbrack_{t_{1}}^{t_{2}} - \\int_{t_{1}}^{t_{2}}{\\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right)\\eta dt} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\frac{\\partial L}{\\partial q} - \\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right) \\right\\rbrack\\eta dt}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c7\/c74\/c7464690e1c4439ba4a7270cfedb35f8.svg\" width=\"683\" height=\"49\" data-width=\"85.342\" data-height=\"6.132\" data-vertical-align=\"-2.5\" style=\"width: 85.342ex; height: 6.132ex; vertical-align: -2.5ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c7\/c74\/c7464690e1c4439ba4a7270cfedb35f8.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c7\/c74\/c7464690e1c4439ba4a7270cfedb35f8.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u043b\u0438 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0434\u043b\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u043e\u0433\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\eta\" alt=\"\\eta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/ffe\/ffe9f913124f345732e9f00fa258552e.svg\" width=\"12\" height=\"18\" data-width=\"1.124\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.124ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/ffe\/ffe9f913124f345732e9f00fa258552e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/ffe\/ffe9f913124f345732e9f00fa258552e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0430 \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u043d\u043e\u0439 \u043b\u0435\u043c\u043c\u0435 \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0433\u043e \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u044d\u0442\u043e \u0437\u043d\u0430\u0447\u0438\u0442, \u0447\u0442\u043e \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u0430\u044f \u0441\u043a\u043e\u0431\u043a\u0430 \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e. \u041f\u043e\u043b\u0443\u0447\u0438\u043c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\frac{\\partial L}{\\partial q} - \\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right) = 0\" alt=\"\\frac{\\partial L}{\\partial q} - \\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right) = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/32e\/32e657d04438a81ec1b99ca0b9c7d470.svg\" width=\"167\" height=\"43\" data-width=\"20.867\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 20.867ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/32e\/32e657d04438a81ec1b99ca0b9c7d470.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/32e\/32e657d04438a81ec1b99ca0b9c7d470.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0435\u0432\u043e\u043c \u0444\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u043c\u0435 \u0432\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u0442\u0430\u043a\u0436\u0435 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u043d\u044b\u0439 \u0438\u043c\u043f\u0443\u043b\u044c\u0441 <img decoding=\"async\" class=\"formula inline\" source=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" alt=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg\" width=\"64\" height=\"42\" data-width=\"7.972\" data-height=\"5.231\" data-vertical-align=\"-2.05\" style=\"width: 7.972ex; height: 5.231ex; vertical-align: -2.05ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041c\u043e\u0436\u043d\u043e \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u0441\u0438\u0441\u0442\u0435\u043c\u0443 \u0438\u0437 \u0448\u043a\u043e\u043b\u044c\u043d\u043e\u0439 \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0438 \u0438 \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0438\u0435 \u0441 \u043f\u0440\u0438\u0432\u044b\u0447\u043d\u044b\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c: <img decoding=\"async\" class=\"formula inline\" source=\"p = \\frac{\\partial}{\\partial v}\\left( \\frac{1}{2}mv^{2} - U(x) \\right) = mv\" alt=\"p = \\frac{\\partial}{\\partial v}\\left( \\frac{1}{2}mv^{2} - U(x) \\right) = mv\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/dbe\/52b\/c2e\/dbe52bc2e51d28b4b3019360cb84300f.svg\" width=\"255\" height=\"45\" data-width=\"30.713\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 30.713ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/dbe\/52b\/c2e\/dbe52bc2e51d28b4b3019360cb84300f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/dbe\/52b\/c2e\/dbe52bc2e51d28b4b3019360cb84300f.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442 \u0441\u0438\u0441\u0442\u0435\u043c\u0443 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u043d\u044b\u0445 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"q\" alt=\"q\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg\" width=\"12\" height=\"12\" data-width=\"1.041\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.041ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u043e\u0432 <img decoding=\"async\" class=\"formula inline\" source=\"p\" alt=\"p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg\" width=\"12\" height=\"12\" data-width=\"1.138\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.138ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0438\u0430\u043d <img decoding=\"async\" class=\"formula inline\" source=\"H = p\\dot{q} - L\" alt=\"H = p\\dot{q} - L\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f37\/f37f4bbbb7eba6b68c0ddab4a385c408.svg\" width=\"92\" height=\"18\" data-width=\"11.511\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 11.511ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f37\/f37f4bbbb7eba6b68c0ddab4a385c408.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f37\/f37f4bbbb7eba6b68c0ddab4a385c408.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0421\u0440\u0430\u0437\u0443 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u043e\u0435 \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u0435: \u044d\u0442\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0443\u0436\u0435 \u043f\u043e\u0445\u043e\u0436\u0435 \u043d\u0430 \u0441\u043e\u0434\u0435\u0440\u0436\u0438\u043c\u043e\u0435 \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c\u0430 \u0432 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0438 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430, \u043d\u043e \u043f\u043e\u0434\u043e\u0439\u0434\u0435\u043c \u043a \u044d\u0442\u043e\u0439 \u0441\u0432\u044f\u0437\u0438 \u043e\u0441\u0442\u043e\u0440\u043e\u0436\u043d\u0435\u0435. \u041f\u043e\u043a\u0430 \u043e\u0442\u043c\u0435\u0442\u0438\u043c: \u0432 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0435 \u0442\u0438\u043f\u0438\u0447\u043d\u043e, \u0447\u0442\u043e \u043a\u0438\u043d\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"T = \\frac{mv^{2}}{2} = \\frac{p\\dot{q}}{2}\" alt=\"T = \\frac{mv^{2}}{2} = \\frac{p\\dot{q}}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c00\/c005f38e4ce761ffef6353e6825d5fec.svg\" width=\"127\" height=\"40\" data-width=\"15.868\" data-height=\"4.968\" data-vertical-align=\"-1.918\" style=\"width: 15.868ex; height: 4.968ex; vertical-align: -1.918ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c00\/c005f38e4ce761ffef6353e6825d5fec.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c00\/c005f38e4ce761ffef6353e6825d5fec.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"H = p\\dot{q} - L = 2T - (T - U) = T + U\" alt=\"H = p\\dot{q} - L = 2T - (T - U) = T + U\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/79\/791\/791b4947da39d414229e290806940d08.svg\" width=\"296\" height=\"18\" data-width=\"36.983\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 36.983ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/79\/791\/791b4947da39d414229e290806940d08.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/79\/791\/791b4947da39d414229e290806940d08.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442.\u0435. \u043f\u043e\u043b\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f.<\/p>\n<p>\u0420\u0430\u0441\u043f\u0438\u0448\u0435\u043c \u0438 \u043f\u0440\u043e\u0432\u0430\u0440\u044c\u0438\u0440\u0443\u0435\u043c \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0435 \u0432 \u0442\u0435\u0440\u043c\u0438\u043d\u0430\u0445 \u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0438\u0430\u043d\u0430 (\u0434\u0432\u0430\u0436\u0434\u044b \u0438 \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e \u043f\u043e q,p):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"S\\lbrack q,p\\rbrack = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack p\\dot{q} - H(q,p) \\right\\rbrack dt},\\ \\ 0 = \\frac{dS}{d\\varepsilon} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\dot{q} - \\frac{\\partial H}{\\partial p} \\right\\rbrack\\xi dt} - \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\dot{p} + \\frac{\\partial H}{\\partial q} \\right\\rbrack\\eta dt}\" alt=\"S\\lbrack q,p\\rbrack = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack p\\dot{q} - H(q,p) \\right\\rbrack dt},\\ \\ 0 = \\frac{dS}{d\\varepsilon} = \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\dot{q} - \\frac{\\partial H}{\\partial p} \\right\\rbrack\\xi dt} - \\int_{t_{1}}^{t_{2}}{\\left\\lbrack \\dot{p} + \\frac{\\partial H}{\\partial q} \\right\\rbrack\\eta dt}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/49\/491\/4910418afa7186035b0ac96fbc204cac.svg\" width=\"635\" height=\"46\" data-width=\"79.329\" data-height=\"5.731\" data-vertical-align=\"-2.3\" style=\"width: 79.329ex; height: 5.731ex; vertical-align: -2.3ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/49\/491\/4910418afa7186035b0ac96fbc204cac.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/49\/491\/4910418afa7186035b0ac96fbc204cac.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0438\u043c \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\dot{q} = \\frac{\\partial H}{\\partial p},\\ \\ \\dot{p} = - \\frac{\\partial H}{\\partial q}\" alt=\"\\dot{q} = \\frac{\\partial H}{\\partial p},\\ \\ \\dot{p} = - \\frac{\\partial H}{\\partial q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/31\/317\/317cf220eb0c88692667d0cd7aa255a1.svg\" width=\"165\" height=\"41\" data-width=\"20.68\" data-height=\"5.138\" data-vertical-align=\"-2.003\" style=\"width: 20.68ex; height: 5.138ex; vertical-align: -2.003ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/31\/317\/317cf220eb0c88692667d0cd7aa255a1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/31\/317\/317cf220eb0c88692667d0cd7aa255a1.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d \u2013 \u044d\u0442\u043e \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432 \u043e\u0431\u0440\u0430\u0437 \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430. \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"H = \\sup_{\\dot{q}}\\left( p\\dot{q} - L \\right)\" alt=\"H = \\sup_{\\dot{q}}\\left( p\\dot{q} - L \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ed\/edc\/edc5b395b917ed8db2fe79eb11919984.svg\" width=\"136\" height=\"35\" data-width=\"17.056\" data-height=\"4.316\" data-vertical-align=\"-1.592\" style=\"width: 17.056ex; height: 4.316ex; vertical-align: -1.592ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ed\/edc\/edc5b395b917ed8db2fe79eb11919984.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ed\/edc\/edc5b395b917ed8db2fe79eb11919984.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435, \u043c\u044b \u0442\u0430\u043a \u0438 \u0432\u0432\u043e\u0434\u0438\u043b\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"H\" alt=\"H\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg\" width=\"16\" height=\"12\" data-width=\"2.009\" data-height=\"1.545\" style=\"width: 2.009ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u044b\u0448\u0435, \u043d\u043e \u043d\u0435 \u0437\u0430\u043c\u0435\u0442\u0438\u043b\u0438, \u043a\u0430\u043a \u043d\u0435\u044f\u0432\u043d\u043e \u0443\u0447\u043b\u0438, \u0447\u0442\u043e \u043d\u0430\u0445\u043e\u0434\u0438\u043c\u0441\u044f \u0432 \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f. \u0414\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e, <\/p>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"0 = \\frac{d}{d\\dot{q}}\\left( p\\dot{q} - L \\right) = p - \\frac{\\partial L}{\\partial\\dot{q}}\" alt=\"0 = \\frac{d}{d\\dot{q}}\\left( p\\dot{q} - L \\right) = p - \\frac{\\partial L}{\\partial\\dot{q}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/3aa\/3fe\/b01\/3aa3feb0198d980982de7f294735233a.svg\" width=\"219\" height=\"43\" data-width=\"26.344\" data-height=\"5.231\" data-vertical-align=\"-2.084\" style=\"width: 26.344ex; height: 5.231ex; vertical-align: -2.084ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/3aa\/3fe\/b01\/3aa3feb0198d980982de7f294735233a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/3aa\/3fe\/b01\/3aa3feb0198d980982de7f294735233a.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" alt=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg\" width=\"64\" height=\"42\" data-width=\"7.972\" data-height=\"5.231\" data-vertical-align=\"-2.05\" style=\"width: 7.972ex; height: 5.231ex; vertical-align: -2.05ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u0430, \u0442\u043e\u043b\u044c\u043a\u043e \u0442\u0435\u043f\u0435\u0440\u044c \u043e\u043d\u043e \u043d\u0435 \u043f\u043e\u0441\u0442\u0443\u043b\u0438\u0440\u0443\u0435\u0442\u0441\u044f, \u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0438\u0437 \u043f\u043e\u0438\u0441\u043a\u0430 \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c\u0430.<\/p>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u043f\u0440\u043e\u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c \u0438 \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"dH = \\dot{q}dp + pd\\dot{q} - \\frac{\\partial L}{\\partial q}dq - \\frac{\\partial L}{\\partial\\dot{q}}d\\dot{q} - \\frac{\\partial L}{\\partial t}dt = \\dot{q}dp - \\frac{\\partial L}{\\partial q}dq - \\frac{\\partial L}{\\partial t}dt + \\left( p - \\frac{\\partial L}{\\partial\\dot{q}} \\right)d\\dot{q} = \\dot{q}dp - \\frac{\\partial L}{\\partial q}dq - \\frac{\\partial L}{\\partial t}dt\" alt=\"dH = \\dot{q}dp + pd\\dot{q} - \\frac{\\partial L}{\\partial q}dq - \\frac{\\partial L}{\\partial\\dot{q}}d\\dot{q} - \\frac{\\partial L}{\\partial t}dt = \\dot{q}dp - \\frac{\\partial L}{\\partial q}dq - \\frac{\\partial L}{\\partial t}dt + \\left( p - \\frac{\\partial L}{\\partial\\dot{q}} \\right)d\\dot{q} = \\dot{q}dp - \\frac{\\partial L}{\\partial q}dq - \\frac{\\partial L}{\\partial t}dt\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/360\/36037c7fa328a6b159881ad0d9c39e2d.svg\" width=\"843\" height=\"43\" data-width=\"105.383\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 105.383ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/360\/36037c7fa328a6b159881ad0d9c39e2d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/360\/36037c7fa328a6b159881ad0d9c39e2d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u0438 \u044d\u0442\u043e\u043c \u0441\u0430\u043c <img decoding=\"async\" class=\"formula inline\" source=\"H = H(q,p,t)\" alt=\"H = H(q,p,t)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/768\/768c47250dd670ae60759132d84e5523.svg\" width=\"110\" height=\"18\" data-width=\"13.803\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.803ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/768\/768c47250dd670ae60759132d84e5523.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/768\/768c47250dd670ae60759132d84e5523.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0421\u043e\u043f\u043e\u0441\u0442\u0430\u0432\u0438\u043c \u0447\u0430\u0441\u0442\u043d\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"dH = \\frac{\\partial H}{\\partial p}dp + \\frac{\\partial H}{\\partial q}dq + \\frac{\\partial H}{\\partial t}dt\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\dot{q} = \\frac{\\partial H}{\\partial p},\\ \\ \\frac{\\partial H}{\\partial q} = - \\frac{\\partial L}{\\partial q}\" alt=\"dH = \\frac{\\partial H}{\\partial p}dp + \\frac{\\partial H}{\\partial q}dq + \\frac{\\partial H}{\\partial t}dt\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\dot{q} = \\frac{\\partial H}{\\partial p},\\ \\ \\frac{\\partial H}{\\partial q} = - \\frac{\\partial L}{\\partial q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/75\/75e\/75e8801daacafa0e4ad9848f2367479d.svg\" width=\"487\" height=\"41\" data-width=\"60.821\" data-height=\"5.138\" data-vertical-align=\"-2.003\" style=\"width: 60.821ex; height: 5.138ex; vertical-align: -2.003ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/75\/75e\/75e8801daacafa0e4ad9848f2367479d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/75\/75e\/75e8801daacafa0e4ad9848f2367479d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0443\u0440-\u0435 \u042d\u0439\u043b\u0435\u0440\u0430-\u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 \u2013 \u043d\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\frac{\\partial L}{\\partial q} = \\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right) = \\dot{p},\\ \\ \u0442.\u0435.\\dot{p} = - \\frac{\\partial H}{\\partial q}\" alt=\"\\frac{\\partial L}{\\partial q} = \\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial\\dot{q}} \\right) = \\dot{p},\\ \\ \u0442.\u0435.\\dot{p} = - \\frac{\\partial H}{\\partial q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77d\/77d49631c986f5b2703f3ae7fcec9f22.svg\" width=\"306\" height=\"43\" data-width=\"38.189\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 38.189ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77d\/77d49631c986f5b2703f3ae7fcec9f22.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77d\/77d49631c986f5b2703f3ae7fcec9f22.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u0438\u0434\u043d\u0430 \u043f\u0435\u0440\u0432\u0430\u044f \u0431\u043e\u043b\u044c\u0448\u0430\u044f \u043f\u043e\u043b\u044c\u0437\u0430 \u043e\u0442 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430: \u043c\u044b \u0441\u0432\u0435\u043b\u0438 \u041e\u0414\u0423 2\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430 \u043a \u043f\u0430\u0440\u0435 \u041e\u0414\u0423 1\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430. \u041f\u043b\u044e\u0441 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442 \u043c\u0435\u0442\u043e\u0434\u044b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u043e\u0449\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u0432 \u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u043e\u0432\u043e\u043c \u0444\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u043c\u0435 (\u0430 \u0432 \u043a\u0432\u0430\u043d\u0442\u043e\u0432\u043e\u0439 \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0435 \u0432 \u043f\u0440\u0438\u043d\u0446\u0438\u043f\u0435 \u0432\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u0438\u043c\u0435\u043d\u043d\u043e \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440 \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0430), \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u0447\u0430\u0441\u0442\u043e.<\/p>\n<p>\u041f\u043e\u0447\u0435\u043c\u0443 \u044d\u0442\u043e \u0432\u043e\u043e\u0431\u0449\u0435 \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442? \u0418 \u043f\u0430\u0440\u0430 \u043a\u043e\u043c\u043c\u0435\u043d\u0442\u0430\u0440\u0438\u0435\u0432<\/p>\n<p>\u041f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043a\u043e\u0440\u0440\u0435\u043a\u0442\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e, \u0435\u0441\u043b\u0438 \u043e\u0442\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q} \\rightarrow p\" alt=\"\\dot{q} \\rightarrow p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4b5\/4b5365c88e0928a1505e455dad7169c4.svg\" width=\"46\" height=\"18\" data-width=\"5.698\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 5.698ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4b5\/4b5365c88e0928a1505e455dad7169c4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4b5\/4b5365c88e0928a1505e455dad7169c4.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043b\u043e\u043a\u0430\u043b\u044c\u043d\u043e \u043e\u0431\u0440\u0430\u0442\u0438\u043c\u043e (\u043c\u044b \u0432\u0435\u0434\u044c \u0432\u044b\u0440\u0430\u0436\u0430\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q}\" alt=\"\\dot{q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg\" width=\"12\" height=\"18\" data-width=\"1.041\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 1.041ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0447\u0435\u0440\u0435\u0437 <img decoding=\"async\" class=\"formula inline\" source=\"p\" alt=\"p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg\" width=\"12\" height=\"12\" data-width=\"1.138\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.138ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u0437 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" alt=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg\" width=\"64\" height=\"42\" data-width=\"7.972\" data-height=\"5.231\" data-vertical-align=\"-2.05\" style=\"width: 7.972ex; height: 5.231ex; vertical-align: -2.05ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u0414\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u2013 \u043d\u0435\u0432\u044b\u0440\u043e\u0436\u0434\u0435\u043d\u043d\u043e\u0441\u0442\u044c \u0433\u0435\u0441\u0441\u0438\u0430\u043d\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}}\" alt=\"\\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/42\/42a\/42a385ba447f378c4580f3d911e8126f.svg\" width=\"39\" height=\"44\" data-width=\"4.922\" data-height=\"5.5\" data-vertical-align=\"-2.184\" style=\"width: 4.922ex; height: 5.5ex; vertical-align: -2.184ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/42\/42a\/42a385ba447f378c4580f3d911e8126f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/42\/42a\/42a385ba447f378c4580f3d911e8126f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043d\u043e \u0431\u043e\u043b\u0435\u0435 \u0444\u0438\u0437\u0438\u0447\u043d\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u2013 \u0442.\u0435. \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u0438 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d\u0430 \u043f\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q}\" alt=\"\\dot{q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg\" width=\"12\" height=\"18\" data-width=\"1.041\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 1.041ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u0447\u0435\u043c\u0443 \u0431\u043e\u043b\u0435\u0435 \u0444\u0438\u0437\u0438\u0447\u043d\u043e, \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u043e\u043c \u043f\u0440\u0438\u043c\u0435\u0440\u0435: <img decoding=\"async\" class=\"formula inline\" source=\"L = \\frac{mv^{2}}{2} - U(x),\\ \\ \\ \\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}} = m &gt; 0\" alt=\"L = \\frac{mv^{2}}{2} - U(x),\\ \\ \\ \\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}} = m &gt; 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/822\/822888294e417fe1c7dfa5bccb0e1d1b.svg\" width=\"272\" height=\"44\" data-width=\"33.956\" data-height=\"5.5\" data-vertical-align=\"-2.184\" style=\"width: 33.956ex; height: 5.5ex; vertical-align: -2.184ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/822\/822888294e417fe1c7dfa5bccb0e1d1b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/822\/822888294e417fe1c7dfa5bccb0e1d1b.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d\u0430 \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442 \u043c\u0430\u0441\u0441\u044b \u0438 \u0438\u0445 \u0430\u043d\u0430\u043b\u043e\u0433\u0438, \u0447\u0442\u043e \u0447\u0430\u0441\u0442\u043e \u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u043c. \u0421\u0440\u0430\u0437\u0443 \u043e\u0442\u043c\u0435\u0442\u0438\u043c: <img decoding=\"async\" class=\"formula inline\" source=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" alt=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg\" width=\"64\" height=\"42\" data-width=\"7.972\" data-height=\"5.231\" data-vertical-align=\"-2.05\" style=\"width: 7.972ex; height: 5.231ex; vertical-align: -2.05ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q} = \\frac{\\partial H}{\\partial p}\" alt=\"\\dot{q} = \\frac{\\partial H}{\\partial p}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5ce45b94321b55bc3793b8f31bb9d9.svg\" width=\"67\" height=\"41\" data-width=\"8.343\" data-height=\"5.138\" data-vertical-align=\"-2.003\" style=\"width: 8.343ex; height: 5.138ex; vertical-align: -2.003ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5ce45b94321b55bc3793b8f31bb9d9.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5ce45b94321b55bc3793b8f31bb9d9.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u0437\u0430\u0438\u043c\u043d\u043e \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0435 \u043e\u0442\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0433\u0435\u0441\u0441\u0438\u0430\u043d\u044b <img decoding=\"async\" class=\"formula inline\" source=\"L\" alt=\"L\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg\" width=\"12\" height=\"12\" data-width=\"1.541\" data-height=\"1.545\" style=\"width: 1.541ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"H\" alt=\"H\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg\" width=\"16\" height=\"12\" data-width=\"2.009\" data-height=\"1.545\" style=\"width: 2.009ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0435 \u043c\u0430\u0442\u0440\u0438\u0446\u044b (\u0438\u0437 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438). \u0412 \u0447\u0430\u0441\u0442\u043d\u043e\u0441\u0442\u0438, \u0433\u0435\u0441\u0441\u0438\u0430\u043d <img decoding=\"async\" class=\"formula inline\" source=\"H\" alt=\"H\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg\" width=\"16\" height=\"12\" data-width=\"2.009\" data-height=\"1.545\" style=\"width: 2.009ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c1\/c1d\/c1d9f50f86825a1a2302ec2449c17196.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b \u0441 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c\u044e \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0445 \u043c\u0430\u0441\u0441 (\u043f\u043e\u0434\u0432\u0438\u0436\u043d\u043e\u0441\u0442\u0435\u0439)<\/p>\n<p>\u0415\u0441\u0442\u044c \u0431\u043e\u043b\u0435\u0435 \u0433\u043b\u0443\u0431\u043e\u043a\u0438\u0439 \u0432\u0437\u0433\u043b\u044f\u0434: \u043f\u043e\u0441\u0442\u0443\u043b\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u0440\u044f\u0434 \u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0439, \u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0432\u044b\u0432\u043e\u0434\u044f\u0442\u0441\u044f \u0441\u0443\u0436\u0434\u0435\u043d\u0438\u044f \u043e \u0432\u0438\u0434\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"L\" alt=\"L\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg\" width=\"12\" height=\"12\" data-width=\"1.541\" data-height=\"1.545\" style=\"width: 1.541ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<ol>\n<li>\n<p>\u043e\u0434\u043d\u043e\u0440\u043e\u0434\u043d\u043e\u0441\u0442\u044c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\rightarrow T\" alt=\"\\rightarrow T\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/368\/368759a048b138d2b1cf323190b6d79d.svg\" width=\"36\" height=\"18\" data-width=\"4.484\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.484ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/368\/368759a048b138d2b1cf323190b6d79d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/368\/368759a048b138d2b1cf323190b6d79d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u043e\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q}\" alt=\"\\dot{q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg\" width=\"12\" height=\"18\" data-width=\"1.041\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 1.041ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p>\u0438\u0437\u043e\u0442\u0440\u043e\u043f\u043d\u043e\u0441\u0442\u044c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\rightarrow T\" alt=\"\\rightarrow T\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/368\/368759a048b138d2b1cf323190b6d79d.svg\" width=\"36\" height=\"18\" data-width=\"4.484\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.484ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/368\/368759a048b138d2b1cf323190b6d79d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/368\/368759a048b138d2b1cf323190b6d79d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u0430\u044f \u0444\u043e\u0440\u043c\u0430 \u043e\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q}\" alt=\"\\dot{q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg\" width=\"12\" height=\"18\" data-width=\"1.041\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 1.041ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p>\u0438\u0437 \u043f\u0440\u0438\u043d\u0446\u0438\u043f\u0430 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438 \u0413\u0430\u043b\u0438\u043b\u0435\u044f \u0441\u043b\u0435\u0434\u0443\u0435\u0442, \u0447\u0442\u043e \u0437\u0430\u043a\u043e\u043d\u044b \u043c\u0435\u0445\u0430\u043d\u0438\u043a\u0438 \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u044b \u0432 \u043b\u044e\u0431\u043e\u0439 \u0418\u0421\u041e, \u0430 \u0437\u043d\u0430\u0447\u0438\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"U\" alt=\"U\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4c\/4c6\/4c614360da93c0a041b22e537de151eb.svg\" width=\"14\" height=\"13\" data-width=\"1.735\" data-height=\"1.595\" data-vertical-align=\"-0.05\" style=\"width: 1.735ex; height: 1.595ex; vertical-align: -0.05ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4c\/4c6\/4c614360da93c0a041b22e537de151eb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4c\/4c6\/4c614360da93c0a041b22e537de151eb.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043c\u043e\u0436\u0435\u0442 \u0437\u0430\u0432\u0438\u0441\u0435\u0442\u044c \u0442\u043e\u043b\u044c\u043a\u043e \u043e\u0442 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"q\" alt=\"q\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg\" width=\"12\" height=\"12\" data-width=\"1.041\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.041ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<\/ol>\n<p>\u041f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f, \u0447\u0442\u043e \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"L\" alt=\"L\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg\" width=\"12\" height=\"12\" data-width=\"1.541\" data-height=\"1.545\" style=\"width: 1.541ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d20\/d20caec3b48a1eef164cb4ca81ba2587.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0447\u0430\u0441\u0442\u043e \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u043d\u0430 \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u043e\u043c \u0443\u0440\u043e\u0432\u043d\u0435. \u041d\u043e \u043d\u0435 \u0432\u0441\u0435\u0433\u0434\u0430: \u0438\u043d\u043e\u0433\u0434\u0430 \u0444\u0438\u0437\u0438\u043a\u0430 \u0434\u043e\u043f\u0443\u0441\u043a\u0430\u0435\u0442 \u043d\u0435\u0432\u044b\u043f\u0443\u043a\u043b\u044b\u0435 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d\u044b, \u0438 \u0442\u043e\u0433\u0434\u0430 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e\u0441\u0442\u044c \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 vs \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0430 \u0442\u0435\u0440\u044f\u0435\u0442\u0441\u044f.<\/p>\n<p>\u0414\u043e\u043f\u043e\u043b\u043d\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0441\u043b\u0443\u0447\u0430\u0439, \u043a\u043e\u0433\u0434\u0430 \u043a\u0430\u043a\u043e\u0439-\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\dot{q}\" alt=\"\\dot{q}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg\" width=\"12\" height=\"18\" data-width=\"1.041\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 1.041ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4d3\/4d33fb4e487aa85e4c53565979df6979.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0435\u043b\u044c\u0437\u044f \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u0447\u0435\u0440\u0435\u0437 \u0441\u0432\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"p\" alt=\"p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg\" width=\"12\" height=\"12\" data-width=\"1.138\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.138ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u0437 <img decoding=\"async\" class=\"formula inline\" source=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" alt=\"p = \\frac{\\partial L}{\\partial\\dot{q}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg\" width=\"64\" height=\"42\" data-width=\"7.972\" data-height=\"5.231\" data-vertical-align=\"-2.05\" style=\"width: 7.972ex; height: 5.231ex; vertical-align: -2.05ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd843334753c78c1550c4bf75d89fd1e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u043a\u0430\u043a \u043c\u044b \u0434\u0435\u043b\u0430\u043b\u0438). \u042d\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{dp}{d\\dot{q}} = \\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}} = 0\" alt=\"\\frac{dp}{d\\dot{q}} = \\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}} = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a8\/3a86fd00f3281d3835bff4af4603bc6f.svg\" width=\"123\" height=\"44\" data-width=\"15.397\" data-height=\"5.5\" data-vertical-align=\"-2.184\" style=\"width: 15.397ex; height: 5.5ex; vertical-align: -2.184ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a8\/3a86fd00f3281d3835bff4af4603bc6f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a8\/3a86fd00f3281d3835bff4af4603bc6f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0414\u0432\u0430\u0436\u0434\u044b \u043f\u0440\u043e\u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0443\u0435\u043c: <img decoding=\"async\" class=\"formula inline\" source=\"L = C_{1}(q) \\cdot \\dot{q} + C_{2}(q)\" alt=\"L = C_{1}(q) \\cdot \\dot{q} + C_{2}(q)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9b\/9bf\/9bf40c62479bea6d03dbd5edb5fdb4b3.svg\" width=\"166\" height=\"18\" data-width=\"20.811\" data-height=\"2.301\" data-vertical-align=\"-0.585\" style=\"width: 20.811ex; height: 2.301ex; vertical-align: -0.585ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9b\/9bf\/9bf40c62479bea6d03dbd5edb5fdb4b3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9b\/9bf\/9bf40c62479bea6d03dbd5edb5fdb4b3.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u043c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"H = \\sup_{\\dot{q}}\\left( p\\dot{q} - L \\right) = \\sup_{\\dot{q}}\\left( \\left( p - C_{1}(q) \\right) \\cdot \\dot{q} - C_{2}(q) \\right) = \\left\\{ \\begin{matrix} - C_{2}(q),\\ \\ p = C_{1}(q) \\\\ \\infty,\\ \\ p \\neq C_{1}(q) \\end{matrix} \\right.\" alt=\"H = \\sup_{\\dot{q}}\\left( p\\dot{q} - L \\right) = \\sup_{\\dot{q}}\\left( \\left( p - C_{1}(q) \\right) \\cdot \\dot{q} - C_{2}(q) \\right) = \\left\\{ \\begin{matrix} - C_{2}(q),\\ \\ p = C_{1}(q) \\\\ \\infty,\\ \\ p \\neq C_{1}(q) \\end{matrix} \\right.\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/640\/640c644735cdcfd09b2e14aaa0d841a7.svg\" width=\"569\" height=\"47\" data-width=\"71.114\" data-height=\"5.861\" data-vertical-align=\"-2.365\" style=\"width: 71.114ex; height: 5.861ex; vertical-align: -2.365ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/640\/640c644735cdcfd09b2e14aaa0d841a7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/640\/640c644735cdcfd09b2e14aaa0d841a7.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u043d\u0430\u0448 \u0441\u0442\u0430\u0440\u044b\u0439 \u043f\u0440\u0438\u043c\u0435\u0440<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) = kx + b\\ \\\u00a0 \\Rightarrow \\ \\ f^{}(p) = \\sup_{x}\\left( (p - k) \\cdot x - b \\right) = \\left\\{ \\begin{matrix} - b,\\ \\ p = k \\\\ \\infty,\\ \\ p \\neq k \\end{matrix} \\right.\" alt=\"f(x) = kx + b\\ \\\u00a0 \\Rightarrow \\ \\ f^{}(p) = \\sup_{x}\\left( (p - k) \\cdot x - b \\right) = \\left\\{ \\begin{matrix} - b,\\ \\ p = k \\\\ \\infty,\\ \\ p \\neq k \\end{matrix} \\right.\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9e\/9e2\/9e23868f3cdb149a538e589f93ffbc12.svg\" width=\"486\" height=\"43\" data-width=\"60.742\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 60.742ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9e\/9e2\/9e23868f3cdb149a538e589f93ffbc12.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9e\/9e2\/9e23868f3cdb149a538e589f93ffbc12.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0438\u0432\u0448\u0438\u0439\u0441\u044f \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u043a\u043e\u043d\u0435\u0447\u043d\u0430 \u043b\u0438\u0448\u044c \u043f\u0440\u0438 \u0441\u043f\u0435\u0446\u0438\u0444\u0438\u0447\u043d\u043e\u043c \u0443\u0441\u043b\u043e\u0432\u0438\u0438 \u043d\u0430 \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u044b, \u0442.\u0435. \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u044b \u043d\u0435 \u043c\u0435\u043d\u044f\u044e\u0442\u0441\u044f \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u043e. \u042d\u0442\u043e \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043f\u0435\u0440\u0432\u0438\u0447\u043d\u043e\u0439 \u0441\u0432\u044f\u0437\u044c\u044e, \u0438 \u0432 \u0441\u0432\u043e\u0435\u0439 \u043e\u0441\u043d\u043e\u0432\u0435 \u2013 \u044d\u0442\u043e \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432 \u043e\u0431\u0440\u0430\u0437 \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439 \u043f\u043e \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0438 \u0447\u0430\u0441\u0442\u0438 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d\u0430. \u041f\u043e \u043d\u0430\u0437\u0432\u0430\u043d\u0438\u044e \u044f\u0441\u043d\u043e, \u0447\u0442\u043e \u0435\u0441\u0442\u044c \u0438 \u0432\u0442\u043e\u0440\u0438\u0447\u043d\u044b\u0435 &#8212; \u043e\u043d\u0438 \u0443\u0441\u0442\u0440\u043e\u0435\u043d\u044b \u0441\u043b\u043e\u0436\u043d\u0435\u0435<\/p>\n<p>\u0417\u0430\u0434\u0430\u0447\u0443 \u043c\u043e\u0436\u043d\u043e \u0441\u0432\u0435\u0441\u0442\u0438 \u043a \u0435\u0449\u0435 \u0431\u043e\u043b\u0435\u0435 \u0443\u0434\u043e\u0431\u043d\u043e\u043c\u0443 \u0432\u0438\u0434\u0443, \u0435\u0441\u043b\u0438 \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u0432 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b, \u0432 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0438\u0430\u043d <img decoding=\"async\" class=\"formula inline\" source=\"\\equiv 0\" alt=\"\\equiv 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cad\/cad217b26c672f75e3bf223785199c59.svg\" width=\"28\" height=\"18\" data-width=\"3.52\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.52ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cad\/cad217b26c672f75e3bf223785199c59.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cad\/cad217b26c672f75e3bf223785199c59.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0420\u0435\u0448\u0435\u043d\u0438\u0435 \u043d\u0430\u043c \u0441\u0435\u0439\u0447\u0430\u0441 \u043d\u0435 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u043e, \u0432 \u0438\u0442\u043e\u0433\u0435 \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u0441\u044f \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0430-\u042f\u043a\u043e\u0431\u0438 (\u043f\u0440\u0438\u0433\u043e\u0434\u0438\u0442\u0441\u044f, \u043d\u043e \u043d\u0435 \u0441\u043a\u043e\u0440\u043e, S &#8212; \u0432\u0434\u043e\u043b\u044c \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u0438):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\frac{\\partial S}{\\partial t} + H\\left( q,\\frac{\\partial S}{\\partial q},t \\right) = 0,\" alt=\"\\frac{\\partial S}{\\partial t} + H\\left( q,\\frac{\\partial S}{\\partial q},t \\right) = 0,\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a7\/a7c\/a7c360e10524f95fcc73db0ab32c467b.svg\" width=\"197\" height=\"43\" data-width=\"24.6\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 24.6ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a7\/a7c\/a7c360e10524f95fcc73db0ab32c467b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a7\/a7c\/a7c360e10524f95fcc73db0ab32c467b.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0418\u0442\u043e\u0433: \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u043f\u0435\u0440\u0435\u043f\u0438\u0441\u0430\u0442\u044c \u0437\u0430\u0434\u0430\u0447\u0443 \u0432 \u0432\u0438\u0434\u0435, \u0432 \u043a\u043e\u0442\u043e\u0440\u043e\u043c \u0435\u0435 \u0438\u043d\u043e\u0433\u0434\u0430 \u043f\u0440\u043e\u0449\u0435 \u0440\u0435\u0448\u0430\u0442\u044c.<\/p>\n<hr\/>\n<h2>\u041e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f<\/h2>\n<p>\u0412 \u044d\u0442\u043e\u043c \u0440\u0430\u0437\u0434\u0435\u043b\u0435 \u043e\u0447\u0435\u043d\u044c \u043c\u043d\u043e\u0433\u043e \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0439 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430, \u0441\u0444\u043e\u043a\u0443\u0441\u0438\u0440\u0443\u0435\u043c\u0441\u044f \u043d\u0430 \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u0438\u0445 \u043a\u043b\u044e\u0447\u0435\u0432\u044b\u0445. \u041a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u0430 \u0432\u0441\u0435\u0433\u043e \u0440\u0430\u0437\u0434\u0435\u043b\u0430 &#8212; \u043d\u0430\u0439\u0442\u0438 \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0437\u0430\u0434\u0430\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438.<\/p>\n<p>\u0427\u0430\u0449\u0435 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0442 \u0432 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u043c \u0432\u0438\u0434\u0435: <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p) = \\sup_{x}\\left( \\left\\langle x,p \\right\\rangle - f(x) \\right)\" alt=\"f^{*}(p) = \\sup_{x}\\left( \\left\\langle x,p \\right\\rangle - f(x) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07c\/07c246e6e508a00902e6537aa7f58c1f.svg\" width=\"209\" height=\"30\" data-width=\"26.074\" data-height=\"3.737\" data-vertical-align=\"-1.303\" style=\"width: 26.074ex; height: 3.737ex; vertical-align: -1.303ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07c\/07c246e6e508a00902e6537aa7f58c1f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07c\/07c246e6e508a00902e6537aa7f58c1f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0414\u043b\u044f \u0433\u043b\u0430\u0434\u043a\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439: <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(p) = \\left\\langle x,p \\right\\rangle - f(x)\" alt=\"f^{*}(p) = \\left\\langle x,p \\right\\rangle - f(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/63\/631\/6315ac10b0281f4ad887b3a5a4553cb5.svg\" width=\"164\" height=\"18\" data-width=\"20.53\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 20.53ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/63\/631\/6315ac10b0281f4ad887b3a5a4553cb5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/63\/631\/6315ac10b0281f4ad887b3a5a4553cb5.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u0440\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"x = (\\nabla f)^{- 1}(p)\" alt=\"x = (\\nabla f)^{- 1}(p)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6c\/6c8\/6c8b861e2cba617c9a2464ee62bce3c8.svg\" width=\"115\" height=\"21\" data-width=\"14.331\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 14.331ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6c\/6c8\/6c8b861e2cba617c9a2464ee62bce3c8.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6c\/6c8\/6c8b861e2cba617c9a2464ee62bce3c8.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e\u0442\u0441\u044e\u0434\u0430 \u0436\u0435 \u0441\u043b\u0435\u0434\u0443\u0435\u0442: <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla f^{*} = (\\nabla f)^{- 1}\" alt=\"\\nabla f^{*} = (\\nabla f)^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e3\/e3b\/e3be862d063af86f310417ecb7d68399.svg\" width=\"115\" height=\"21\" data-width=\"14.375\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 14.375ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e3\/e3b\/e3be862d063af86f310417ecb7d68399.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e3\/e3b\/e3be862d063af86f310417ecb7d68399.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla^{2}f^{*} = \\left( \\nabla^{2}f \\right)^{- 1}\" alt=\"\\nabla^{2}f^{*} = \\left( \\nabla^{2}f \\right)^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/51\/512\/51215259130b6962c42f99fb1121e755.svg\" width=\"133\" height=\"26\" data-width=\"16.663\" data-height=\"3.239\" data-vertical-align=\"-1.054\" style=\"width: 16.663ex; height: 3.239ex; vertical-align: -1.054ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/51\/512\/51215259130b6962c42f99fb1121e755.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/51\/512\/51215259130b6962c42f99fb1121e755.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u0435\u0441\u043b\u0438 \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u043e\u0431\u0440\u0430\u0442\u0438\u043c &#8212; \u043f\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0435 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0447\u0442\u043e \u043c\u044b \u043f\u0440\u043e\u043d\u0430\u0431\u043b\u044e\u0434\u0430\u043b\u0438 \u043d\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0435 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d\u0430 \u0438 \u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0438\u0430\u043d\u0430)<\/p>\n<p>\u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u043a\u043b\u044e\u0447\u0435\u0432\u0443\u044e \u0438\u0434\u0435\u044e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430: \u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0435\u0441\u0442\u044c \u0432\u0435\u0440\u0445\u043d\u044f\u044f \u043e\u0433\u0438\u0431\u0430\u044e\u0449\u0430\u044f \u0441\u0432\u043e\u0438\u0445 \u043d\u0438\u0436\u043d\u0438\u0445 \u043e\u0446\u0435\u043d\u043e\u043a. \u0412 \u043c\u0435\u0442\u043e\u0434\u0430\u0445 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \u0447\u0430\u0441\u0442\u043e \u0443\u0434\u043e\u0431\u043d\u043e \u043d\u0435 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0430\u0442\u044c \u043e \u043c\u0438\u043d\u0438\u043c\u0443\u043c\u0435 \u0441\u0430\u043c\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0430 \u0434\u0430\u0432\u0430\u0442\u044c \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0438 \u043d\u0430 \u0435\u0435 \u043d\u0438\u0436\u043d\u0438\u0435 \u043e\u0446\u0435\u043d\u043a\u0438, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0442\u0430\u043a\u0430\u044f \u0442\u0440\u0430\u043a\u0442\u043e\u0432\u043a\u0430 \u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043f\u043e\u043b\u0435\u0437\u043d\u0430.<\/p>\n<p>\u0423\u0441\u043b\u043e\u0432\u043d\u0430\u044f \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f<\/p>\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0437\u0430\u0434\u0430\u0447\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\min_{x}{f(x)}\" alt=\"\\min_{x}{f(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/104\/104321fd0785169ff430db14e123a0ce.svg\" width=\"68\" height=\"26\" data-width=\"8.447\" data-height=\"3.298\" data-vertical-align=\"-1.083\" style=\"width: 8.447ex; height: 3.298ex; vertical-align: -1.083ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/104\/104321fd0785169ff430db14e123a0ce.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/104\/104321fd0785169ff430db14e123a0ce.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u0440\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"Ax = b\" alt=\"Ax = b\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fef\/fefb19183bff83a05ea8a98b793adab3.svg\" width=\"56\" height=\"14\" data-width=\"6.979\" data-height=\"1.805\" data-vertical-align=\"-0.186\" style=\"width: 6.979ex; height: 1.805ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fef\/fefb19183bff83a05ea8a98b793adab3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fef\/fefb19183bff83a05ea8a98b793adab3.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0421\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d (\u0441\u0440\u0430\u0437\u0443 \u043e\u0442\u043c\u0435\u0442\u0438\u043c: \u044d\u0442\u043e \u043d\u0435 \u0442\u043e\u0442 \u0436\u0435 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0431\u044b\u043b \u0432 \u0444\u0438\u0437\u0438\u043a\u0435, \u043a \u0441\u043e\u0436\u0430\u043b\u0435\u043d\u0438\u044e, \u0442\u0443\u0442 \u043a\u043e\u043b\u043b\u0438\u0437\u0438\u044f \u0438\u043c\u0435\u043d \u2013 \u043b\u0438\u0448\u044c \u0441\u043e\u0432\u043f\u0430\u0434\u0435\u043d\u0438\u0435):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"L(x,\\lambda) = f(x) + \\lambda^{T}(b - Ax)\" alt=\"L(x,\\lambda) = f(x) + \\lambda^{T}(b - Ax)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdd\/fddf1633b3f4317dfa9b0e301bcb46a9.svg\" width=\"225\" height=\"21\" data-width=\"28.122\" data-height=\"2.583\" data-vertical-align=\"-0.726\" style=\"width: 28.122ex; height: 2.583ex; vertical-align: -0.726ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdd\/fddf1633b3f4317dfa9b0e301bcb46a9.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdd\/fddf1633b3f4317dfa9b0e301bcb46a9.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0427\u0442\u043e \u043f\u0440\u043e\u0438\u0437\u043e\u0448\u043b\u043e: \u0434\u043b\u044f \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0445 x (\u0442.\u0435. \u0442\u0430\u043a\u0438\u0445, \u0447\u0442\u043e \u0432\u0442\u043e\u0440\u043e\u0435 \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u043e\u0435 \u0437\u0430\u043d\u0443\u043b\u044f\u0435\u0442\u0441\u044f) \u043c\u043e\u0436\u043d\u043e \u0432\u0437\u044f\u0442\u044c \u043b\u044e\u0431\u043e\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda\" alt=\"\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg\" width=\"12\" height=\"18\" data-width=\"1.319\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.319ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0432\u0441\u0435 \u0440\u0430\u0432\u043d\u043e \u0432 \u0441\u043a\u043e\u0431\u043a\u0435 \u043d\u043e\u043b\u044c. \u041d\u043e \u0442\u0435\u043f\u0435\u0440\u044c \u0434\u0430\u0432\u0430\u0439\u0442\u0435 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043d\u0435 \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435 x, \u0430 \u0432\u043e\u043e\u0431\u0449\u0435 \u0432\u0441\u0435 x: \u0441\u0432\u043e\u0431\u043e\u0434\u044b \u0431\u043e\u043b\u044c\u0448\u0435, \u0437\u043d\u0430\u0447\u0438\u0442 \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0442\u043e\u043b\u044c\u043a\u043e \u043e\u043f\u0443\u0441\u0442\u0438\u0442\u0441\u044f:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\inf_{\u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435\\ x}{f(x)} = \\inf_{\u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435\\ x}\\left( f(x) + \\lambda^{T}(b - Ax) \\right) \\geq \\inf_{x}\\left( f(x) + \\lambda^{T}(b - Ax) \\right)\" alt=\"\\inf_{\u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435\\ x}{f(x)} = \\inf_{\u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0435\\ x}\\left( f(x) + \\lambda^{T}(b - Ax) \\right) \\geq \\inf_{x}\\left( f(x) + \\lambda^{T}(b - Ax) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/643\/6430c670060a5a0728593fbfc8f7a5aa.svg\" width=\"612\" height=\"33\" data-width=\"76.535\" data-height=\"4.141\" data-vertical-align=\"-1.505\" style=\"width: 76.535ex; height: 4.141ex; vertical-align: -1.505ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/643\/6430c670060a5a0728593fbfc8f7a5aa.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/643\/6430c670060a5a0728593fbfc8f7a5aa.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u0431\u043e\u0437\u043d\u0430\u0447\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"g(\\lambda) = \\inf_{x}\\left( f(x) + \\lambda^{T}(b - Ax) \\right)\" alt=\"g(\\lambda) = \\inf_{x}\\left( f(x) + \\lambda^{T}(b - Ax) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c3d\/c3d89233ea1191caf464b3954ab6e891.svg\" width=\"244\" height=\"29\" data-width=\"30.538\" data-height=\"3.619\" data-vertical-align=\"-1.244\" style=\"width: 30.538ex; height: 3.619ex; vertical-align: -1.244ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c3d\/c3d89233ea1191caf464b3954ab6e891.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c3d\/c3d89233ea1191caf464b3954ab6e891.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u0441\u0435\u043c\u0435\u0439\u0441\u0442\u0432\u043e \u043e\u0446\u0435\u043d\u043e\u043a \u0441\u043d\u0438\u0437\u0443 \u043d\u0430 \u043d\u0430\u0448\u0443 \u0444\u0443\u043d\u043a\u0446\u0438\u044e f. \u0418\u0437 \u044d\u0442\u043e\u0433\u043e \u0441\u0435\u043c\u0435\u0439\u0441\u0442\u0432\u0430 \u043b\u043e\u0433\u0438\u0447\u043d\u043e \u0432\u0437\u044f\u0442\u044c \u043b\u0443\u0447\u0448\u0443\u044e: <img decoding=\"async\" class=\"formula inline\" source=\"\\sup_{\\lambda}{g(\\lambda)}\" alt=\"\\sup_{\\lambda}{g(\\lambda)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c5c\/c5c34baef6b99ab73623aa0084c90daa.svg\" width=\"64\" height=\"31\" data-width=\"7.943\" data-height=\"3.869\" data-vertical-align=\"-1.369\" style=\"width: 7.943ex; height: 3.869ex; vertical-align: -1.369ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c5c\/c5c34baef6b99ab73623aa0084c90daa.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c5c\/c5c34baef6b99ab73623aa0084c90daa.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u0432\u0441\u0435 \u0435\u0449\u0435 \u043e\u0446\u0435\u043d\u043a\u0430 \u0441\u043d\u0438\u0437\u0443, \u043d\u043e \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u0430\u044f (\u043d\u0430\u0438\u0431\u043e\u043b\u0435\u0435 \u00ab\u0442\u0443\u0433\u0430\u044f\u00bb).<\/p>\n<p>\u041f\u0435\u0440\u0435\u0439\u0434\u044f \u043a \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0435, \u043c\u044b \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u043f\u0440\u043e\u0432\u0435\u043b\u0438 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0432\u0441\u0435\u0433\u0434\u0430 \u0431\u044b\u043b\u043e \u0437\u0430\u0434\u0430\u0447\u0435\u0439 \u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0439 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \u0441 \u043b\u0438\u043d\u0435\u0439\u043d\u044b\u043c\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f\u043c\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"g(\\lambda) = \\inf_{x}{L(x,\\lambda)} = \\lambda^{T}b + \\inf_{x}\\left\\lbrack f(x) - \\left( A^{T}\\lambda \\right)^{T}x \\right\\rbrack = \\lambda^{T}b - \\sup_{x}\\left\\lbrack \\left( A^{T}\\lambda \\right)^{T}x - f(x) \\right\\rbrack = \\lambda^{T}b - f^{*}\\left( A^{T}\\lambda \\right)\" alt=\"g(\\lambda) = \\inf_{x}{L(x,\\lambda)} = \\lambda^{T}b + \\inf_{x}\\left\\lbrack f(x) - \\left( A^{T}\\lambda \\right)^{T}x \\right\\rbrack = \\lambda^{T}b - \\sup_{x}\\left\\lbrack \\left( A^{T}\\lambda \\right)^{T}x - f(x) \\right\\rbrack = \\lambda^{T}b - f^{*}\\left( A^{T}\\lambda \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d5\/d51\/d517579abdfffdd252ad71652f5d0647.svg\" width=\"782\" height=\"37\" data-width=\"97.803\" data-height=\"4.641\" data-vertical-align=\"-1.755\" style=\"width: 97.803ex; height: 4.641ex; vertical-align: -1.755ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d5\/d51\/d517579abdfffdd252ad71652f5d0647.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d5\/d51\/d517579abdfffdd252ad71652f5d0647.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0432\u0441\u0435 \u0432\u0440\u0435\u043c\u044f <img decoding=\"async\" class=\"formula inline\" source=\"g(\\lambda)\" alt=\"g(\\lambda)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fa\/fa8\/fa86ac1b31aa3dcdc2140ca1e5bfa92b.svg\" width=\"33\" height=\"18\" data-width=\"4.158\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.158ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fa\/fa8\/fa86ac1b31aa3dcdc2140ca1e5bfa92b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fa\/fa8\/fa86ac1b31aa3dcdc2140ca1e5bfa92b.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044e \u0431\u044b\u043b\u0438 \u043e\u0446\u0435\u043d\u043a\u0430\u043c\u0438 \u0441\u043d\u0438\u0437\u0443, <img decoding=\"async\" class=\"formula inline\" source=\"\\sup_{\\lambda}g \\leq \\inf f\" alt=\"\\sup_{\\lambda}g \\leq \\inf f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/23\/23f\/23fa588c7f3923af2b216aa28a984799.svg\" width=\"98\" height=\"31\" data-width=\"12.231\" data-height=\"3.869\" data-vertical-align=\"-1.369\" style=\"width: 12.231ex; height: 3.869ex; vertical-align: -1.369ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/23\/23f\/23fa588c7f3923af2b216aa28a984799.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/23\/23f\/23fa588c7f3923af2b216aa28a984799.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u0440\u0438\u0447\u0435\u043c \u0432 \u043d\u0435\u0432\u044b\u043f\u0443\u043a\u043b\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0437\u0430\u0437\u043e\u0440 \u043e\u0436\u0438\u0434\u0430\u0435\u0442\u0441\u044f, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u0430 \u0432\u043e\u0441\u0441\u0442\u0430\u043d\u0430\u0432\u043b\u0438\u0432\u0430\u0435\u0442 \u043b\u0438\u0448\u044c <img decoding=\"async\" class=\"formula inline\" source=\"f^{**} = conv\\ f\" alt=\"f^{**} = conv\\ f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/868\/8687af4a623c5f7a5412c481f10f5fb4.svg\" width=\"100\" height=\"18\" data-width=\"12.511\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 12.511ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/868\/8687af4a623c5f7a5412c481f10f5fb4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/868\/8687af4a623c5f7a5412c481f10f5fb4.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a<\/p>\n<p>\u0427\u0430\u0441\u0442\u043e \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043d\u0435\u043b\u044c\u0437\u044f\/\u0441\u043b\u043e\u0436\u043d\u043e\/\u0434\u043e\u043b\u0433\u043e \u0438\u0441\u043a\u0430\u0442\u044c \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438, \u043b\u0438\u0431\u043e \u043d\u0430\u0441 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u0443\u0435\u0442 \u043b\u044e\u0431\u043e\u0439 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0445\u043e\u0440\u043e\u0448\u0438\u0439 \u043b\u043e\u043a\u0430\u043b\u044c\u043d\u044b\u0439 \u043c\u0438\u043d\u0438\u043c\u0443\u043c. \u0422\u043e\u0433\u0434\u0430 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u044e\u0442 \u0438\u0442\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u044b\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u044b, \u0432 \u0447\u0430\u0441\u0442\u043d\u043e\u0441\u0442\u0438, \u043f\u043e\u043f\u0443\u043b\u044f\u0440\u043d\u044b\u0439 \u0432 ML \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a. \u041a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u0438\u0434\u0435\u044f \u2013 \u043f\u043e\u0434\u0433\u043e\u043d\u044f\u0435\u043c \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u0443 \u0441\u0442\u043e\u0440\u043e\u043d\u0443, \u043a\u0443\u0434\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0443\u0431\u044b\u0432\u0430\u0435\u0442:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"x_{k + 1} = x_{k} - \\eta \\cdot \\nabla f\\left( x_{k} \\right)\" alt=\"x_{k + 1} = x_{k} - \\eta \\cdot \\nabla f\\left( x_{k} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c3b\/c3b6af3b938d9ff82dbe45058a2a2d11.svg\" width=\"182\" height=\"18\" data-width=\"22.799\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 22.799ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c3b\/c3b6af3b938d9ff82dbe45058a2a2d11.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c3\/c3b\/c3b6af3b938d9ff82dbe45058a2a2d11.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u0443 \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u043d\u0443\u044e \u0438\u043d\u0442\u0443\u0438\u0446\u0438\u044e \u043c\u043e\u0436\u043d\u043e \u043e\u0431\u043e\u0441\u043d\u043e\u0432\u0430\u0442\u044c \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438. \u0410\u043f\u043f\u0440\u043e\u043a\u0441\u0438\u043c\u0438\u0440\u0443\u0435\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044e f \u0435\u0435 \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0432 \u0442\u0435\u043a\u0443\u0449\u0435\u0439 \u0442\u043e\u0447\u043a\u0435. \u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u0432 \u0435\u0435 \u043c\u0438\u043d\u0438\u043c\u0443\u043c, \u043d\u043e \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439, \u044f\u0441\u043d\u043e, \u043d\u0430 \u0431\u0435\u0441\u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0441\u0442\u0438, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0434\u043e\u0431\u0430\u0432\u0438\u043c \u0448\u0442\u0440\u0430\u0444 \u0437\u0430 \u043e\u0442\u0445\u043e\u0434 \u043e\u0442 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0439 \u0441\u0442\u0440\u043e\u0447\u043a\u0438 \u2013 \u043d\u043e\u0440\u043c\u0443 \u0448\u0430\u0433\u0430 \u0441 \u0440\u0435\u0433\u0443\u043b\u0438\u0440\u0443\u0435\u043c\u044b\u043c \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u043e\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\eta\" alt=\"\\eta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/ffe\/ffe9f913124f345732e9f00fa258552e.svg\" width=\"12\" height=\"18\" data-width=\"1.124\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.124ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/ffe\/ffe9f913124f345732e9f00fa258552e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/ffe\/ffe9f913124f345732e9f00fa258552e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) \\approx f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle\" alt=\"f(x) \\approx f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d25\/d254695d0d954f88b950ffa3a27d3cbc.svg\" width=\"265\" height=\"18\" data-width=\"33.067\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 33.067ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d25\/d254695d0d954f88b950ffa3a27d3cbc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d25\/d254695d0d954f88b950ffa3a27d3cbc.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"x_{k + 1} = \\arg{\\min\\left\\lbrack f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle + \\frac{1}{2\\eta}\\left\\| x - x_{k} \\right\\|^{2} \\right\\rbrack} = \\arg{\\min_{x}\\left\\lbrack \\eta\\left\\langle \\nabla f\\left( x_{k} \\right),x \\right\\rangle + \\frac{1}{2}\\left\\| x - x_{k} \\right\\|^{2} \\right\\rbrack}\" alt=\"x_{k + 1} = \\arg{\\min\\left\\lbrack f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle + \\frac{1}{2\\eta}\\left\\| x - x_{k} \\right\\|^{2} \\right\\rbrack} = \\arg{\\min_{x}\\left\\lbrack \\eta\\left\\langle \\nabla f\\left( x_{k} \\right),x \\right\\rangle + \\frac{1}{2}\\left\\| x - x_{k} \\right\\|^{2} \\right\\rbrack}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7d\/7d9\/7d91a685fcdf23cf77340ce0c814ca48.svg\" width=\"797\" height=\"43\" data-width=\"99.577\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 99.577ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7d\/7d9\/7d91a685fcdf23cf77340ce0c814ca48.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7d\/7d9\/7d91a685fcdf23cf77340ce0c814ca48.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u0438\u0434\u043d\u043e, \u0447\u0442\u043e \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u0435\u043d \u0437\u0430\u0434\u0430\u0447\u0435 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \u0441 \u0435\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u044b\u043c \u0448\u0442\u0440\u0430\u0444\u043e\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\left\\| x - x_{k} \\right\\|^{2}\" alt=\"\\left\\| x - x_{k} \\right\\|^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ef\/ef2\/ef20aaf25d6cbbd00121791e0a316749.svg\" width=\"77\" height=\"22\" data-width=\"9.625\" data-height=\"2.71\" data-vertical-align=\"-0.79\" style=\"width: 9.625ex; height: 2.71ex; vertical-align: -0.79ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ef\/ef2\/ef20aaf25d6cbbd00121791e0a316749.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ef\/ef2\/ef20aaf25d6cbbd00121791e0a316749.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u043e\u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043f\u043e\u0434 \u0430\u0440\u0433\u043c\u0438\u043d\u0438\u043c\u0443\u043c\u043e\u043c \u043f\u043e x:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\nabla f\\left( x_{k} \\right) + \\frac{1}{\\eta}\\left( x - x_{k} \\right) = 0\\ \\\u00a0 \\Rightarrow \\ \\ x_{k + 1} = x_{k} - \\eta \\cdot \\nabla f\\left( x_{k} \\right)\" alt=\"\\nabla f\\left( x_{k} \\right) + \\frac{1}{\\eta}\\left( x - x_{k} \\right) = 0\\ \\\u00a0 \\Rightarrow \\ \\ x_{k + 1} = x_{k} - \\eta \\cdot \\nabla f\\left( x_{k} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ad\/ad9\/ad99d9ec19585669caaa77f00bb34d64.svg\" width=\"427\" height=\"41\" data-width=\"53.338\" data-height=\"5.077\" data-vertical-align=\"-1.973\" style=\"width: 53.338ex; height: 5.077ex; vertical-align: -1.973ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ad\/ad9\/ad99d9ec19585669caaa77f00bb34d64.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ad\/ad9\/ad99d9ec19585669caaa77f00bb34d64.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u043e \u0437\u0434\u0435\u0441\u044c \u0437\u0430\u043b\u043e\u0436\u0435\u043d\u043e \u0434\u043e\u0432\u043e\u043b\u044c\u043d\u043e \u0442\u043e\u043d\u043a\u043e\u0435 \u0434\u043e\u043f\u0443\u0449\u0435\u043d\u0438\u0435, \u0434\u043b\u044f \u043f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u044f \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u043d\u0443\u0436\u043d\u043e \u043f\u043e\u0433\u0440\u0443\u0437\u0438\u0442\u044c\u0441\u044f \u0432 \u043f\u0440\u0438\u0440\u043e\u0434\u0443 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u044b\u0445 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432.<\/p>\n<p>\u0421\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0435 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e \u2014 \u044d\u0442\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0432\u0441\u0435\u0445 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b\u0445 \u043b\u0438\u043d\u0435\u0439\u043d\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u043e\u043d\u0430\u043b\u043e\u0432, \u0437\u0430\u0434\u0430\u043d\u043d\u044b\u0445 \u043d\u0430 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435. \u041f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u0438\u043b\u043e \u043d\u0430\u0441 \u0438\u043c\u0435\u043d\u043d\u043e \u0442\u0443\u0434\u0430. \u0412 \u043e\u0431\u043e\u0438\u0445 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430\u0445 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043c\u0435\u0442\u0440\u0438\u043a\u0430 &#8212; \u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0430\u044f \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"M\" alt=\"M\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/696\/69691c7bdcc3ce6d5d8a1361f22d04ac.svg\" width=\"19\" height=\"12\" data-width=\"2.378\" data-height=\"1.545\" style=\"width: 2.378ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/696\/69691c7bdcc3ce6d5d8a1361f22d04ac.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/696\/69691c7bdcc3ce6d5d8a1361f22d04ac.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 \u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u0432\u0432\u043e\u0434\u0438\u0442\u0441\u044f \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435: <img decoding=\"async\" class=\"formula inline\" source=\"\\left\\langle u,v \\right\\rangle_{M} = u^{T}Mv\" alt=\"\\left\\langle u,v \\right\\rangle_{M} = u^{T}Mv\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/15c\/15c649d65435535f5a9a4e5e897acb1d.svg\" width=\"129\" height=\"21\" data-width=\"16.127\" data-height=\"2.663\" data-vertical-align=\"-0.766\" style=\"width: 16.127ex; height: 2.663ex; vertical-align: -0.766ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/15c\/15c649d65435535f5a9a4e5e897acb1d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/15c\/15c649d65435535f5a9a4e5e897acb1d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 \u0436\u0438\u0432\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\partial f\" alt=\"\\partial f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/61\/61d\/61d235736e622497a52726f602397caf.svg\" width=\"20\" height=\"18\" data-width=\"2.525\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.525ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/61\/61d\/61d235736e622497a52726f602397caf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/61\/61d\/61d235736e622497a52726f602397caf.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u201c\u0441\u044b\u0440\u043e\u0439\u201d \u043a\u043e\u0432\u0435\u043a\u0442\u043e\u0440-\u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442, \u0430 \u0432 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u043c &#8212; <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla f\" alt=\"\\nabla f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b8\/8b89a43d93ebcb54fa54d24afb9e51b8.svg\" width=\"25\" height=\"18\" data-width=\"3.129\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.129ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b8\/8b89a43d93ebcb54fa54d24afb9e51b8.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b8\/8b89a43d93ebcb54fa54d24afb9e51b8.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u0440\u0438\u0447\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\left\\langle \\nabla f,v \\right\\rangle_{M} = \\partial f(v)\" alt=\"\\left\\langle \\nabla f,v \\right\\rangle_{M} = \\partial f(v)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/91\/911\/91170470e8af7b1d7615ae2475a98492.svg\" width=\"138\" height=\"19\" data-width=\"17.261\" data-height=\"2.342\" data-vertical-align=\"-0.606\" style=\"width: 17.261ex; height: 2.342ex; vertical-align: -0.606ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/91\/911\/91170470e8af7b1d7615ae2475a98492.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/91\/911\/91170470e8af7b1d7615ae2475a98492.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0438\u043b\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla f = M^{- 1}\\partial f\" alt=\"\\nabla f = M^{- 1}\\partial f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd1\/bd1f32fdedc98db1fc0e49d95d1206b1.svg\" width=\"107\" height=\"21\" data-width=\"13.403\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 13.403ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd1\/bd1f32fdedc98db1fc0e49d95d1206b1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd1\/bd1f32fdedc98db1fc0e49d95d1206b1.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412 \u0415\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u043e\u0439 \u043c\u0435\u0442\u0440\u0438\u043a\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"M = I\" alt=\"M = I\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/af\/af6\/af67ed35d58fa0b971387d623691b256.svg\" width=\"52\" height=\"14\" data-width=\"6.535\" data-height=\"1.731\" data-vertical-align=\"-0.186\" style=\"width: 6.535ex; height: 1.731ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/af\/af6\/af67ed35d58fa0b971387d623691b256.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/af\/af6\/af67ed35d58fa0b971387d623691b256.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla f = \\partial f\" alt=\"\\nabla f = \\partial f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2a\/2ae\/2ae63d9421ac7904c8c0ba88e8ad5fbf.svg\" width=\"69\" height=\"18\" data-width=\"8.671\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.671ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2a\/2ae\/2ae63d9421ac7904c8c0ba88e8ad5fbf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2a\/2ae\/2ae63d9421ac7904c8c0ba88e8ad5fbf.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043a\u0430\u043a \u043c\u044b \u0438 \u043f\u0438\u0441\u0430\u043b\u0438 \u0432 \u0444\u043e\u0440\u043c\u0443\u043b\u0435 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u043e\u0433\u043e \u0441\u043f\u0443\u0441\u043a\u0430.<\/p>\n<p>\u041c\u044b \u0440\u0430\u043d\u0435\u0435 \u0443\u0436\u0435 \u0441\u0442\u0430\u043b\u043a\u0438\u0432\u0430\u043b\u0438\u0441\u044c \u0441 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u0435\u0439, \u043a\u043e\u0433\u0434\u0430 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u0438\u0435 \u043d\u0435 \u0438\u0437\u043c\u0435\u043d\u044f\u0435\u0442 \u0444\u0443\u043d\u043a\u0446\u0438\u044e: <img decoding=\"async\" class=\"formula inline\" source=\"\\psi = \\frac{1}{2}x^{2}\" alt=\"\\psi = \\frac{1}{2}x^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/84\/84b\/84b0606a3b2586570a074374fd21706b.svg\" width=\"71\" height=\"37\" data-width=\"8.898\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 8.898ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/84\/84b\/84b0606a3b2586570a074374fd21706b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/84\/84b\/84b0606a3b2586570a074374fd21706b.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043a \u043d\u0435\u0439, \u0430 \u0442\u043e\u0447\u043d\u0435\u0435, \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\psi(x) = \\frac{1}{2}x^{T}Mx\" alt=\"\\psi(x) = \\frac{1}{2}x^{T}Mx\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cf\/cf7\/cf765c2a54b87bc0f9bf80d70e5be77f.svg\" width=\"128\" height=\"37\" data-width=\"15.951\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 15.951ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cf\/cf7\/cf765c2a54b87bc0f9bf80d70e5be77f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cf\/cf7\/cf765c2a54b87bc0f9bf80d70e5be77f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0422\u043e\u0433\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla\\psi(x) = Mx,\\ \\ \\ \\nabla\\psi^{*}(p) = M^{- 1}p\" alt=\"\\nabla\\psi(x) = Mx,\\ \\ \\ \\nabla\\psi^{*}(p) = M^{- 1}p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/99\/99f\/99f9ffdc6416db18890803a5121d4e8c.svg\" width=\"255\" height=\"21\" data-width=\"31.935\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 31.935ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/99\/99f\/99f9ffdc6416db18890803a5121d4e8c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/99\/99f\/99f9ffdc6416db18890803a5121d4e8c.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u043d\u0430\u0448\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla\\psi^{*} = (\\nabla\\psi)^{- 1}\" alt=\"\\nabla\\psi^{*} = (\\nabla\\psi)^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1b\/1bd\/1bd6d5729958f80b5a809cd44c913898.svg\" width=\"118\" height=\"21\" data-width=\"14.712\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 14.712ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1b\/1bd\/1bd6d5729958f80b5a809cd44c913898.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1b\/1bd\/1bd6d5729958f80b5a809cd44c913898.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u0437 \u043d\u0430\u0447\u0430\u043b\u0430 \u0440\u0430\u0437\u0434\u0435\u043b\u0430. \u041a\u0430\u043a \u0432\u0438\u0434\u043d\u043e, \u0441\u0430\u043c\u0430 \u043c\u0430\u0442\u0440\u0438\u0446\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"M = \\nabla^{2}\\psi\" alt=\"M = \\nabla^{2}\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/253\/253b63685e88af6853b9cddbe797fba0.svg\" width=\"78\" height=\"21\" data-width=\"9.74\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 9.74ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/253\/253b63685e88af6853b9cddbe797fba0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/253\/253b63685e88af6853b9cddbe797fba0.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412 \u0444\u0438\u0437\u0438\u043a\u0435 \u0443 \u043d\u0430\u0441 \u0443\u0436\u0435 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u043b \u043e\u0431\u0440\u0430\u0442\u0438\u043c\u044b\u0439 \u0433\u0435\u0441\u0441\u0438\u0430\u043d <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}} = m &gt; 0\" alt=\"\\frac{\\partial^{2}L}{\\partial{\\dot{q}}^{2}} = m &gt; 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f64\/f645e40b90641a7951dcb8436bd15a21.svg\" width=\"113\" height=\"44\" data-width=\"14.073\" data-height=\"5.5\" data-vertical-align=\"-2.184\" style=\"width: 14.073ex; height: 5.5ex; vertical-align: -2.184ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f64\/f645e40b90641a7951dcb8436bd15a21.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f64\/f645e40b90641a7951dcb8436bd15a21.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043c\u0430\u0441\u0441\u0430\/\u043c\u0430\u0442\u0440\u0438\u0446\u0430 \u0438\u043d\u0435\u0440\u0446\u0438\u0438. \u042d\u0442\u043e \u0442\u0430 \u0441\u0430\u043c\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"M\" alt=\"M\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/696\/69691c7bdcc3ce6d5d8a1361f22d04ac.svg\" width=\"19\" height=\"12\" data-width=\"2.378\" data-height=\"1.545\" style=\"width: 2.378ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/696\/69691c7bdcc3ce6d5d8a1361f22d04ac.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/69\/696\/69691c7bdcc3ce6d5d8a1361f22d04ac.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u0440\u043e\u0441\u0442\u043e \u0432 \u0434\u0440\u0443\u0433\u043e\u043c \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435.<\/p>\n<p>\u0412\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u043c\u0441\u044f \u043a \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u043e\u043c\u0443 \u0441\u043f\u0443\u0441\u043a\u0443. \u0412 \u0431\u043e\u043b\u0435\u0435 \u043e\u0431\u0449\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"M \\neq I\" alt=\"M \\neq I\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/203\/2035f3cc0e0b6bfdc7421f0333b530be.svg\" width=\"52\" height=\"18\" data-width=\"6.535\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.535ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/203\/2035f3cc0e0b6bfdc7421f0333b530be.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/203\/2035f3cc0e0b6bfdc7421f0333b530be.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0431\u0443\u0434\u0435\u0442:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"x_{k + 1} = x_{k} - \\eta \\cdot \\nabla f\\left( x_{k} \\right) = x_{k} - \\eta \\cdot \\nabla\\psi^{*}(\\partial f)\" alt=\"x_{k + 1} = x_{k} - \\eta \\cdot \\nabla f\\left( x_{k} \\right) = x_{k} - \\eta \\cdot \\nabla\\psi^{*}(\\partial f)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/301\/301ff5d457d610dedb79143c1a80c7bf.svg\" width=\"338\" height=\"18\" data-width=\"42.286\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 42.286ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/301\/301ff5d457d610dedb79143c1a80c7bf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/301\/301ff5d457d610dedb79143c1a80c7bf.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u043e\u0441\u0442\u043e \u043c\u044b \u043d\u0435 \u0437\u0430\u043c\u0435\u0442\u0438\u043b\u0438, \u043a\u0430\u043a \u043d\u0435\u044f\u0432\u043d\u043e \u0432\u044b\u0431\u0440\u0430\u043b\u0438 \u0441\u0430\u043c\u043e\u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u0443\u044e <img decoding=\"async\" class=\"formula inline\" source=\"\\psi\" alt=\"\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg\" width=\"12\" height=\"18\" data-width=\"1.473\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.473ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0442\u0443 \u0441\u0430\u043c\u0443\u044e \u0438\u0437 \u043d\u0430\u0447\u0430\u043b\u0430 \u0441\u0442\u0430\u0442\u044c\u0438.<\/p>\n<p>\u0417\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a<\/p>\n<p>\u0421\u0443\u043c\u043c\u0438\u0440\u0443\u0435\u043c, \u0432 \u0447\u0435\u043c \u0431\u044b\u043b\u0430 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430. \u041c\u044b \u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0435\u043c \u043e\u0431\u044a\u0435\u043a\u0442 \u0438\u0437 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 (x) \u0441 \u043e\u0431\u044a\u0435\u043a\u0442\u043e\u043c \u0438\u0437 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0433\u043e (\u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442). \u041c\u044b \u043d\u0435\u0437\u0430\u043c\u0435\u0442\u043d\u043e \u043e\u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u0438\u043b\u0438 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"X\" alt=\"X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg\" width=\"15\" height=\"12\" data-width=\"1.928\" data-height=\"1.545\" style=\"width: 1.928ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"X^{*}\" alt=\"X^{*}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f2\/f2d\/f2d6c5f9b59bae5bc8a1ff91f8ae7ded.svg\" width=\"24\" height=\"13\" data-width=\"3.031\" data-height=\"1.678\" style=\"width: 3.031ex; height: 1.678ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f2\/f2d\/f2d6c5f9b59bae5bc8a1ff91f8ae7ded.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f2\/f2d\/f2d6c5f9b59bae5bc8a1ff91f8ae7ded.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0445\u043e\u0442\u044f \u0432 \u043e\u0431\u0449\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u043e\u043d\u0438 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u044b. \u041d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u044d\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e \u043c\u044b \u043d\u0435 \u0443\u0447\u043b\u0438 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044e \u0437\u0430\u0434\u0430\u0447\u0438. \u041a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043f\u0440\u0438\u043c\u0435\u0440 \u2013 \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f \u043d\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u043d\u043e\u043c \u0441\u0438\u043c\u043f\u043b\u0435\u043a\u0441\u0435: <img decoding=\"async\" class=\"formula inline\" source=\"x_{i} \\geq 0,\\ \\ \\sum_{}^{}x_{i} = 1\" alt=\"x_{i} \\geq 0,\\ \\ \\sum_{}^{}x_{i} = 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7bc\/7bc6064ce26bfa15de523263720b45d1.svg\" width=\"148\" height=\"43\" data-width=\"18.523\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 18.523ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7bc\/7bc6064ce26bfa15de523263720b45d1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7bc\/7bc6064ce26bfa15de523263720b45d1.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0415\u0432\u043a\u043b\u0438\u0434\u043e\u0432 \u0448\u0442\u0440\u0430\u0444 \u043d\u0430 \u043d\u0435\u043c \u0432\u0441\u0451 \u0432\u0440\u0435\u043c\u044f \u0443\u0442\u044b\u043a\u0430\u0435\u0442\u0441\u044f \u0432 \u0433\u0440\u0430\u043d\u0438\u0446\u044b, \u0442\u0430\u043a \u0447\u0442\u043e \u0445\u043e\u0442\u0435\u043b\u043e\u0441\u044c \u0431\u044b \u0437\u0430\u043b\u043e\u0436\u0438\u0442\u044c \u0432 \u043c\u0435\u0442\u043e\u0434 \u0443\u0447\u0435\u0442 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432<\/p>\n<p>\u042d\u0442\u0443 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0443 \u0440\u0435\u0448\u0430\u0435\u0442 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0441\u043f\u0443\u0441\u043a\u0430:<\/p>\n<ol>\n<li>\n<p>\u0431\u0435\u0440\u0435\u043c \u0441\u0442\u0440\u043e\u0433\u043e \u0432\u044b\u043f\u0443\u043a\u043b\u0443\u044e \u0433\u043b\u0430\u0434\u043a\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"\\psi\" alt=\"\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg\" width=\"12\" height=\"18\" data-width=\"1.473\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.473ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (distance-generating function, \u00ab\u0437\u0435\u0440\u043a\u0430\u043b\u043e\u00bb)<\/p>\n<\/li>\n<li>\n<p>\u043f\u0435\u0440\u0435\u0432\u043e\u0434\u0438\u043c \u0442\u043e\u0447\u043a\u0443 \u0432 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0435 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e<\/p>\n<\/li>\n<li>\n<p>\u0434\u0435\u043b\u0430\u0435\u043c \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u0432 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 \u2013 \u0442\u0430\u043c, \u0433\u0434\u0435 \u043d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u0436\u0438\u0432\u0443\u0442 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u044b<\/p>\n<\/li>\n<li>\n<p>\u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u043c\u0441\u044f \u0432 \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0435 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e<\/p>\n<\/li>\n<\/ol>\n<p>\u0418\u043c\u0435\u043d\u043d\u043e \u0437\u0430 \u0441\u0447\u0435\u0442 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u043e\u0432 \u043c\u0435\u0436\u0434\u0443 \u0438\u0441\u0445\u043e\u0434\u043d\u044b\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e\u043c \u0438 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u044b\u043c \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u043f\u043e\u043b\u0443\u0447\u0438\u043b \u043d\u0430\u0437\u0432\u0430\u043d\u0438\u0435 \u00ab\u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u044b\u0439\u00bb \u0441\u043f\u0443\u0441\u043a:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"x_{k + 1} = \\nabla\\psi^{*}\\left( \\nabla\\psi\\left( x_{k} \\right) - \\eta \\cdot \\nabla f\\left( x_{k} \\right) \\right)\" alt=\"x_{k + 1} = \\nabla\\psi^{*}\\left( \\nabla\\psi\\left( x_{k} \\right) - \\eta \\cdot \\nabla f\\left( x_{k} \\right) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/46\/46d\/46d97d0106984bf8d3e86759dea6a8fd.svg\" width=\"278\" height=\"18\" data-width=\"34.776\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 34.776ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/46\/46d\/46d97d0106984bf8d3e86759dea6a8fd.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/46\/46d\/46d97d0106984bf8d3e86759dea6a8fd.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<figure class=\"\"><img decoding=\"async\" src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/8af\/882\/094\/8af882094012d52ae6cc6801e66ca4f5.png\" width=\"411\" height=\"392\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/8af\/882\/094\/8af882094012d52ae6cc6801e66ca4f5.png 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/8af\/882\/094\/8af882094012d52ae6cc6801e66ca4f5.png 781w\" loading=\"lazy\" decode=\"async\"\/><\/figure>\n<p>\u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u2013 \u0447\u0430\u0441\u0442\u043d\u044b\u0439 \u0441\u043b\u0443\u0447\u0430\u0439 \u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043f\u0440\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\psi(x) = \\frac{1}{2}x^{T}x\" alt=\"\\psi(x) = \\frac{1}{2}x^{T}x\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/60\/607\/60739d288cc8526069b318a59f1078ec.svg\" width=\"109\" height=\"37\" data-width=\"13.573\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 13.573ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/60\/607\/60739d288cc8526069b318a59f1078ec.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/60\/607\/60739d288cc8526069b318a59f1078ec.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0423 \u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0441\u043f\u0443\u0441\u043a\u0430 \u0435\u0441\u0442\u044c \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c (\u043c\u043e\u0436\u043d\u043e \u0443\u0431\u0435\u0434\u0438\u0442\u044c\u0441\u044f, \u0432\u0437\u044f\u0432 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u043f\u043e\u0434 argmax):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"x_{k + 1} = \\arg{\\min_{x}\\left\\lbrack \\eta\\left\\langle \\nabla f\\left( x_{k} \\right),x \\right\\rangle + D_{\\psi}\\left( x,x_{k} \\right) \\right\\rbrack},\\ \\ \u0433\u0434\u0435\\ D_{\\psi}(x,y) = \\psi(x) - \\psi(y) - \\left\\langle \\nabla\\psi(y),x - y \\right\\rangle\" alt=\"x_{k + 1} = \\arg{\\min_{x}\\left\\lbrack \\eta\\left\\langle \\nabla f\\left( x_{k} \\right),x \\right\\rangle + D_{\\psi}\\left( x,x_{k} \\right) \\right\\rbrack},\\ \\ \u0433\u0434\u0435\\ D_{\\psi}(x,y) = \\psi(x) - \\psi(y) - \\left\\langle \\nabla\\psi(y),x - y \\right\\rangle\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6b\/c6bf5a2f22af6e083f1eab9f7b76e187.svg\" width=\"710\" height=\"26\" data-width=\"88.699\" data-height=\"3.298\" data-vertical-align=\"-1.083\" style=\"width: 88.699ex; height: 3.298ex; vertical-align: -1.083ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6b\/c6bf5a2f22af6e083f1eab9f7b76e187.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6b\/c6bf5a2f22af6e083f1eab9f7b76e187.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0424\u0443\u043d\u043a\u0446\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"D_{\\psi}\" alt=\"D_{\\psi}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/18\/184\/1843914356757e278165c8de2c3e1f2b.svg\" width=\"25\" height=\"19\" data-width=\"3.103\" data-height=\"2.364\" data-vertical-align=\"-0.616\" style=\"width: 3.103ex; height: 2.364ex; vertical-align: -0.616ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/18\/184\/1843914356757e278165c8de2c3e1f2b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/18\/184\/1843914356757e278165c8de2c3e1f2b.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0434\u0438\u0432\u0435\u0440\u0433\u0435\u043d\u0446\u0438\u0435\u0439 \u0411\u0440\u044d\u0433\u043c\u0430\u043d\u0430 \u2013 \u043f\u043e \u0441\u0443\u0442\u0438, \u044d\u0442\u043e \u0430\u043d\u0430\u043b\u043e\u0433 \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f. \u0414\u043b\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\psi = \\frac{1}{2}\\left\\| \\\u00a0 \\cdot \\\u00a0 \\right\\|^{2}\" alt=\"\\psi = \\frac{1}{2}\\left\\| \\\u00a0 \\cdot \\\u00a0 \\right\\|^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/56\/562\/562d81fdb29661323dfa78c9810144eb.svg\" width=\"111\" height=\"37\" data-width=\"13.89\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 13.89ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/56\/562\/562d81fdb29661323dfa78c9810144eb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/56\/562\/562d81fdb29661323dfa78c9810144eb.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0431\u0443\u0434\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"D_{\\psi}(x,y) = \\frac{1}{2}\\left\\| x - y \\right\\|^{2}\" alt=\"D_{\\psi}(x,y) = \\frac{1}{2}\\left\\| x - y \\right\\|^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d11\/d1148a0f71112a8f93ed51d56e23c360.svg\" width=\"175\" height=\"37\" data-width=\"21.834\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 21.834ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d11\/d1148a0f71112a8f93ed51d56e23c360.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d11\/d1148a0f71112a8f93ed51d56e23c360.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0438 \u0432\u0441\u044f \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f \u0441\u0445\u043b\u043e\u043f\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0432 \u043e\u0431\u044b\u0447\u043d\u044b\u0439 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a.<\/p>\n<p>\u0412\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043a \u043f\u0440\u0438\u043c\u0435\u0440\u0443 \u0441 \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0435\u0439 \u043d\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u043d\u043e\u043c \u0441\u0438\u043c\u043f\u043b\u0435\u043a\u0441\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"x_{i} \\geq 0,\\ \\ \\sum_{}^{}x_{i} = 1\" alt=\"x_{i} \\geq 0,\\ \\ \\sum_{}^{}x_{i} = 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7bc\/7bc6064ce26bfa15de523263720b45d1.svg\" width=\"148\" height=\"43\" data-width=\"18.523\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 18.523ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7bc\/7bc6064ce26bfa15de523263720b45d1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7bc\/7bc6064ce26bfa15de523263720b45d1.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041c\u044b \u0443\u043f\u043e\u043c\u0438\u043d\u0430\u043b\u0438, \u0447\u0442\u043e \u0435\u0432\u043a\u043b\u0438\u0434\u043e\u0432 \u0448\u0442\u0440\u0430\u0444 \u043d\u0435 \u0443\u0447\u0438\u0442\u044b\u0432\u0430\u0435\u0442 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430, \u043d\u043e \u043a\u0430\u043a\u043e\u0439 \u043f\u043e\u0434\u043e\u0448\u0435\u043b \u0431\u044b \u043b\u0443\u0447\u0448\u0435? \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u0437\u0435\u0440\u043a\u0430\u043b\u0430 \u043b\u0443\u0447\u0448\u0435 \u0432\u0441\u0435\u0433\u043e \u043f\u043e\u0434\u0445\u043e\u0434\u0438\u0442 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\psi = \\sum_{}^{}{x_{i}\\log x_{i}}\" alt=\"\\psi = \\sum_{}^{}{x_{i}\\log x_{i}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/87\/87d\/87d9a399e2c6447066025250c7c8e055.svg\" width=\"127\" height=\"43\" data-width=\"15.847\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 15.847ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/87\/87d\/87d9a399e2c6447066025250c7c8e055.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/87\/87d\/87d9a399e2c6447066025250c7c8e055.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"D_{\\psi}(x,y) = \\psi(x) - \\psi(y) - \\left\\langle \\nabla\\psi(y),x - y \\right\\rangle = \\sum_{}^{}{x_{i}\\log x_{i}} - \\sum_{}^{}{y_{i}\\log y_{i}} - \\sum_{}^{}{\\left( \\log y_{i} + 1 \\right)\\left( x_{i} - y_{i} \\right)} = \\sum_{}^{}{x_{i}\\log x_{i}} - \\sum_{}^{}{x_{i}\\log y_{i}} - \\sum_{}^{}x_{i} + \\sum_{}^{}y_{i} = \\sum_{}^{}{x_{i}\\log\\frac{x_{i}}{y_{i}}} = KL\\left( x||y \\right)\" alt=\"D_{\\psi}(x,y) = \\psi(x) - \\psi(y) - \\left\\langle \\nabla\\psi(y),x - y \\right\\rangle = \\sum_{}^{}{x_{i}\\log x_{i}} - \\sum_{}^{}{y_{i}\\log y_{i}} - \\sum_{}^{}{\\left( \\log y_{i} + 1 \\right)\\left( x_{i} - y_{i} \\right)} = \\sum_{}^{}{x_{i}\\log x_{i}} - \\sum_{}^{}{x_{i}\\log y_{i}} - \\sum_{}^{}x_{i} + \\sum_{}^{}y_{i} = \\sum_{}^{}{x_{i}\\log\\frac{x_{i}}{y_{i}}} = KL\\left( x||y \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/440\/440d59d768a91e8d83c3fe5af58d3f81.svg\" width=\"1334\" height=\"43\" data-width=\"166.695\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 166.695ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/440\/440d59d768a91e8d83c3fe5af58d3f81.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/440\/440d59d768a91e8d83c3fe5af58d3f81.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0434\u0438\u0432\u0435\u0440\u0433\u0435\u043d\u0446\u0438\u044f \u0411\u0440\u044d\u0433\u043c\u0430\u043d\u0430 \u0434\u043b\u044f \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u0441\u044f KL-\u0434\u0438\u0432\u0435\u0440\u0433\u0435\u043d\u0446\u0438\u0435\u0439 \u2013 \u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u043c \u0430\u043d\u0430\u043b\u043e\u0433\u043e\u043c \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u0434\u043b\u044f \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0439.<\/p>\n<p>\u041c\u0435\u0442\u043e\u0434 \u041d\u044c\u044e\u0442\u043e\u043d\u0430<\/p>\n<p>\u041b\u0438\u043d\u0435\u0439\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u043e\u0447\u0435\u043d\u044c \u0433\u0440\u0443\u0431\u043e\u0435, \u0438 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u0438\u0434\u0435\u044f \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0430\u0442\u044c \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u043e. \u041a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u0438\u0434\u0435\u044f: \u043d\u0430 \u043a\u0430\u0436\u0434\u043e\u043c \u0448\u0430\u0433\u0435 \u0441\u0442\u0440\u043e\u0438\u043c \u043a\u0430\u0441\u0430\u0442\u0435\u043b\u044c\u043d\u0443\u044e \u043f\u0430\u0440\u0430\u0431\u043e\u043b\u0443, \u0438 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0438\u043c \u0432 \u0435\u0435 \u043c\u0438\u043d\u0438\u043c\u0443\u043c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(x) \\approx f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle + \\frac{1}{2}\\left( \\ x - x_{k} \\right)^{T}\\nabla^{2}f\\left( x_{k} \\right)\\left( \\ x - x_{k} \\right)\" alt=\"f(x) \\approx f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle + \\frac{1}{2}\\left( \\ x - x_{k} \\right)^{T}\\nabla^{2}f\\left( x_{k} \\right)\\left( \\ x - x_{k} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ef\/efc\/efccb0ef50535adb343222259532bf07.svg\" width=\"525\" height=\"37\" data-width=\"65.622\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 65.622ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ef\/efc\/efccb0ef50535adb343222259532bf07.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ef\/efc\/efccb0ef50535adb343222259532bf07.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"x_{k + 1} = \\arg{\\min{\\left\\lbrack f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle + \\frac{1}{2}\\left( \\ x - x_{k} \\right)^{T}\\nabla^{2}f\\left( x_{k} \\right)\\left( \\ x - x_{k} \\right) \\right\\rbrack = x_{k} - \\left\\lbrack \\nabla^{2}f\\left( x_{k} \\right) \\right\\rbrack^{- 1}\\partial f}}\" alt=\"x_{k + 1} = \\arg{\\min{\\left\\lbrack f\\left( x_{k} \\right) + \\left\\langle \\nabla f\\left( x_{k} \\right),\\ x - x_{k} \\right\\rangle + \\frac{1}{2}\\left( \\ x - x_{k} \\right)^{T}\\nabla^{2}f\\left( x_{k} \\right)\\left( \\ x - x_{k} \\right) \\right\\rbrack = x_{k} - \\left\\lbrack \\nabla^{2}f\\left( x_{k} \\right) \\right\\rbrack^{- 1}\\partial f}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/99\/991\/991dcfb986031de38d914d4caace6d4a.svg\" width=\"792\" height=\"43\" data-width=\"99.059\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 99.059ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/99\/991\/991dcfb986031de38d914d4caace6d4a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/99\/991\/991dcfb986031de38d914d4caace6d4a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<figure class=\"\"><img decoding=\"async\" src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/d7a\/b9b\/43d\/d7ab9b43d732ba02c9796b94d1cd548d.png\" width=\"508\" height=\"362\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/d7a\/b9b\/43d\/d7ab9b43d732ba02c9796b94d1cd548d.png 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/d7a\/b9b\/43d\/d7ab9b43d732ba02c9796b94d1cd548d.png 781w\" loading=\"lazy\" decode=\"async\"\/><\/figure>\n<p>\u042f\u0441\u043d\u043e, \u0447\u0442\u043e \u0435\u0441\u043b\u0438 \u0441\u0430\u043c\u0430 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u043c\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u0430\u044f, \u0442\u043e \u043e\u043d\u0430 \u0441\u043e\u0432\u043f\u0430\u0434\u0435\u0442 \u0441\u043e \u0441\u0432\u043e\u0438\u043c \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u044b\u043c \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435\u043c, \u0438 \u043c\u044b \u043f\u0435\u0440\u0435\u0439\u0434\u0435\u043c \u0432 \u043e\u043f\u0442\u0438\u043c\u0443\u043c \u0432\u0441\u0435\u0433\u043e \u0437\u0430 1 \u0448\u0430\u0433.<\/p>\n<p>\u0410\u043b\u044c\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432\u043d\u043e \u043c\u0435\u0442\u043e\u0434 \u041d\u044c\u044e\u0442\u043e\u043d\u0430 \u0432\u044b\u0432\u043e\u0434\u044f\u0442 \u0447\u0435\u0440\u0435\u0437 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u043f\u043e\u0438\u0441\u043a\u0430 \u043d\u0443\u043b\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0438 \u0432\u043c\u0435\u0441\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0442 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u0438\u0441\u043a\u043e\u043c\u043e\u0439. \u041d\u0430\u0433\u043b\u044f\u0434\u043d\u043e, \u043d\u043e \u043d\u0430\u043c \u0441\u0435\u0439\u0447\u0430\u0441 \u043d\u0435 \u0442\u0430\u043a \u0432\u0430\u0436\u043d\u043e.<\/p>\n<p>\u0412\u0438\u0434\u043d\u043e, \u0447\u0442\u043e \u043c\u0435\u0442\u043e\u0434 \u041d\u044c\u044e\u0442\u043e\u043d\u0430 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442 \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043c\u0435\u0442\u0440\u0438\u043a\u0438 \u0433\u0435\u0441\u0441\u0438\u0430\u043d <img decoding=\"async\" class=\"formula inline\" source=\"M = \\nabla^{2}f\" alt=\"M = \\nabla^{2}f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fc\/4fc6104c058b513925477ced46587923.svg\" width=\"76\" height=\"21\" data-width=\"9.512\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 9.512ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fc\/4fc6104c058b513925477ced46587923.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fc\/4fc6104c058b513925477ced46587923.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u0435\u043d \u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u043e\u043c\u0443 \u0441\u043f\u0443\u0441\u043a\u0443 \u0441 \u0437\u0435\u0440\u043a\u0430\u043b\u043e\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\psi(x) = \\frac{1}{2}x^{T}\\nabla^{2}f\\left( x_{k} \\right)x\" alt=\"\\psi(x) = \\frac{1}{2}x^{T}\\nabla^{2}f\\left( x_{k} \\right)x\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d4d\/d4dc8da0143353112320d7b63e0eb3db.svg\" width=\"177\" height=\"37\" data-width=\"22.142\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 22.142ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d4d\/d4dc8da0143353112320d7b63e0eb3db.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d4d\/d4dc8da0143353112320d7b63e0eb3db.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u0440\u0438 \u044d\u0442\u043e\u043c \u043f\u0440\u0438 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0435 \u0438\u0437 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 \u043e\u0431\u0440\u0430\u0442\u043d\u043e \u043a \u0438\u043a\u0441\u0430\u043c, \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u0434\u043e\u043c\u043d\u043e\u0436\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u0440\u0430\u0437 \u043d\u0430 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0439 \u0433\u0435\u0441\u0441\u0438\u0430\u043d (\u043a\u0430\u043a \u043c\u044b \u0440\u0430\u043d\u0435\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u043b\u0438 \u0441\u0432\u044f\u0437\u044c \u043c\u0435\u0436\u0434\u0443 \u0433\u0435\u0441\u0441\u0438\u0430\u043d\u043e\u043c \u043f\u0440\u044f\u043c\u043e\u0439 \u0438 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u0435\u0441\u043b\u0438 \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u043d\u0435\u0432\u044b\u0440\u043e\u0436\u0434\u0435\u043d)<\/p>\n<p>\u041c\u0435\u0442\u0440\u0438\u043a\u0443\/\u0433\u0435\u0441\u0441\u0438\u0430\u043d <img decoding=\"async\" class=\"formula inline\" source=\"M = \\nabla^{2}f\" alt=\"M = \\nabla^{2}f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fc\/4fc6104c058b513925477ced46587923.svg\" width=\"76\" height=\"21\" data-width=\"9.512\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 9.512ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fc\/4fc6104c058b513925477ced46587923.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fc\/4fc6104c058b513925477ced46587923.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043c\u043e\u0436\u043d\u043e \u0430\u043f\u043f\u0440\u043e\u043a\u0441\u0438\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c. \u0422\u0430\u043a\u0438\u0435 \u043c\u0435\u0442\u043e\u0434\u044b \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u043a\u0432\u0430\u0437\u0438\u043d\u044c\u044e\u0442\u043e\u043d\u043e\u0432\u0441\u043a\u0438\u043c\u0438. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0435\u0441\u043b\u0438 \u041c \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0430\u0435\u0442\u0441\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0435\u0439 \u0440\u0430\u043d\u0433\u0430 2, \u043c\u0435\u0442\u043e\u0434 \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f BFGS.<\/p>\n<p>\u0418\u0442\u043e\u0433: \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u043a \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0435, \u0433\u0434\u0435 \u0443\u0434\u043e\u0431\u043d\u043e \u0441\u0442\u0440\u043e\u0438\u0442\u044c \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u044b\u0435 \u043e\u0446\u0435\u043d\u043a\u0438 \u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438\u043b\u0438 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 \u0431\u043e\u043b\u0435\u0435 \u043d\u0430\u0433\u043b\u044f\u0434\u043d\u0430.<\/p>\n<hr\/>\n<h2>\u0421\u0442\u0430\u0442\u0444\u0438\u0437 \/ \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u043a\u0430<\/h2>\n<p>\u0412\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043d\u0435\u043c\u043d\u043e\u0433\u043e \u043a \u0444\u0438\u0437\u0438\u043a\u0435. \u041e\u0431\u0441\u0443\u0434\u0438\u043c \u043e\u0434\u043d\u043e \u043b\u043e\u0436\u043d\u043e\u0435 \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u0435, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u044f \u0441\u043b\u044b\u0448\u0430\u043b \u0432 \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435 \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u043a\u0438: \u201c\u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043c\u0435\u043d\u044f\u0435\u0442 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u044b\u0435 \u0438 \u044d\u043a\u0441\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u044b\u0435 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b\u201d.<\/p>\n<p>\u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u00ab\u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u044b\u0435\u00bb \u0438 \u00ab\u044d\u043a\u0441\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u044b\u0435\u00bb? \u041f\u0443\u0441\u0442\u044c \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0443\u0432\u0435\u043b\u0438\u0447\u0438\u043b\u0430\u0441\u044c \u0432 <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda\" alt=\"\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg\" width=\"12\" height=\"18\" data-width=\"1.319\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.319ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0440\u0430\u0437. \u0422\u043e\u0433\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"g(\\lambda X) = \\lambda g(X)\" alt=\"g(\\lambda X) = \\lambda g(X)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b6\/b69\/b69033c159f0f02024afa2f4019b5002.svg\" width=\"122\" height=\"18\" data-width=\"15.189\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 15.189ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b6\/b69\/b69033c159f0f02024afa2f4019b5002.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b6\/b69\/b69033c159f0f02024afa2f4019b5002.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u043a\u0441\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u0430\u044f (\u043e\u0431\u044a\u0435\u043c V, \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f S), \u0430 <img decoding=\"async\" class=\"formula inline\" source=\"g(\\lambda X) = g(X)\" alt=\"g(\\lambda X) = g(X)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a6\/a64\/a641b6234d69387a8a0dd31c6035804b.svg\" width=\"111\" height=\"18\" data-width=\"13.87\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.87ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a6\/a64\/a641b6234d69387a8a0dd31c6035804b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a6\/a64\/a641b6234d69387a8a0dd31c6035804b.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u0430\u044f (\u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430 T, \u0434\u0430\u0432\u043b\u0435\u043d\u0438\u0435 P). \u0412\u043e\u043e\u0431\u0449\u0435, \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0441\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e\u043c <img decoding=\"async\" class=\"formula inline\" source=\"g(\\lambda X) = \\lambda^{k}g(X)\" alt=\"g(\\lambda X) = \\lambda^{k}g(X)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ad\/ad5\/ad594e762515f9b065e389df33e1f91a.svg\" width=\"130\" height=\"21\" data-width=\"16.21\" data-height=\"2.61\" data-vertical-align=\"-0.74\" style=\"width: 16.21ex; height: 2.61ex; vertical-align: -0.74ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ad\/ad5\/ad594e762515f9b065e389df33e1f91a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ad\/ad5\/ad594e762515f9b065e389df33e1f91a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442\u0441\u044f \u043e\u0434\u043d\u043e\u0440\u043e\u0434\u043d\u044b\u043c\u0438 \u043f\u043e\u0440\u044f\u0434\u043a\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"k\" alt=\"k\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg\" width=\"12\" height=\"13\" data-width=\"1.179\" data-height=\"1.595\" data-vertical-align=\"-0.025\" style=\"width: 1.179ex; height: 1.595ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0422\u0430\u043a\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0443\u0434\u043e\u0431\u043d\u044b \u0442\u0435\u043c, \u0447\u0442\u043e \u043c\u0430\u0441\u0448\u0442\u0430\u0431\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u043f\u0440\u0435\u0434\u0441\u043a\u0430\u0437\u0443\u0435\u043c\u043e.<\/p>\n<p>\u0412\u043e\u0437\u044c\u043c\u0435\u043c \u0432\u043d\u0443\u0442\u0440\u0435\u043d\u043d\u044e\u044e \u044d\u043d\u0435\u0440\u0433\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"U(S,V,N)\" alt=\"U(S,V,N)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/61\/613\/613b90f9238164279cd01183fe7e6a32.svg\" width=\"86\" height=\"18\" data-width=\"10.716\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 10.716ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/61\/613\/613b90f9238164279cd01183fe7e6a32.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/61\/613\/613b90f9238164279cd01183fe7e6a32.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0415\u0435 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"S,V,N\" alt=\"S,V,N\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/01\/01e\/01eb22be9b8bc92d497ce7efd51e5a6b.svg\" width=\"58\" height=\"16\" data-width=\"7.22\" data-height=\"2.034\" data-vertical-align=\"-0.439\" style=\"width: 7.22ex; height: 2.034ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/01\/01e\/01eb22be9b8bc92d497ce7efd51e5a6b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/01\/01e\/01eb22be9b8bc92d497ce7efd51e5a6b.svg 781w\" loading=\"lazy\" decode=\"async\"\/> &#8212; \u044d\u043a\u0441\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u044b\u0435, \u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\partial U}{\\partial S} = T,\\ \\ \\frac{\\partial U}{\\partial V} = - P,\\ \\ \\frac{\\partial U}{\\partial N} = \\mu\" alt=\"\\frac{\\partial U}{\\partial S} = T,\\ \\ \\frac{\\partial U}{\\partial V} = - P,\\ \\ \\frac{\\partial U}{\\partial N} = \\mu\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c2d\/c2d6a92f6666d4f8035014f32596f048.svg\" width=\"256\" height=\"38\" data-width=\"32.054\" data-height=\"4.749\" data-vertical-align=\"-1.809\" style=\"width: 32.054ex; height: 4.749ex; vertical-align: -1.809ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c2d\/c2d6a92f6666d4f8035014f32596f048.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c2d\/c2d6a92f6666d4f8035014f32596f048.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u044b\u0435 (\u0438\u0445 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u2013 \u0447\u0438\u0441\u0442\u043e \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043c\u043e\u043c\u0435\u043d\u0442). \u041f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0434\u0430\u0435\u0442 \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u0443\u044e \u044d\u043d\u0435\u0440\u0433\u0438\u044e \u0413\u0435\u043b\u044c\u043c\u0433\u043e\u043b\u044c\u0446\u0430: <img decoding=\"async\" class=\"formula inline\" source=\"F(T,V,N) = \\inf_{S}(U - TS)\" alt=\"F(T,V,N) = \\inf_{S}(U - TS)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/af\/af7\/af7cd0356b4e90a12e0fc941e118acff.svg\" width=\"207\" height=\"28\" data-width=\"25.867\" data-height=\"3.464\" data-vertical-align=\"-1.166\" style=\"width: 25.867ex; height: 3.464ex; vertical-align: -1.166ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/af\/af7\/af7cd0356b4e90a12e0fc941e118acff.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/af\/af7\/af7cd0356b4e90a12e0fc941e118acff.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0438 \u044d\u043a\u0441\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u0430\u044f S \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \u043d\u0430 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u0443\u044e T. \u041d\u043e \u0432\u043e\u0437\u044c\u043c\u0435\u043c \u0442\u0435\u043f\u0435\u0440\u044c \u0432\u0441\u0435 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u044b\u0435 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b \u0432 \u0440\u0430\u0441\u0447\u0435\u0442\u0435 \u043d\u0430 1 \u0447\u0430\u0441\u0442\u0438\u0446\u0443: <img decoding=\"async\" class=\"formula inline\" source=\"U = Nu(s,v),\\ \\ F = Nf(T,v)\" alt=\"U = Nu(s,v),\\ \\ F = Nf(T,v)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d6\/6d691a6b1d13b0c05ad2564bb56ca840.svg\" width=\"228\" height=\"18\" data-width=\"28.539\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 28.539ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d6\/6d691a6b1d13b0c05ad2564bb56ca840.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d6\/6d691a6b1d13b0c05ad2564bb56ca840.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412\u0430\u0436\u043d\u043e, \u0447\u0442\u043e s \u0441\u0442\u0430\u043b\u0430 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u043e\u0439. \u0422\u043e\u0433\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"f(T,v) = \\inf_{s}\\left( u(s,v) - Ts \\right)\" alt=\"f(T,v) = \\inf_{s}\\left( u(s,v) - Ts \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d4b\/d4b7e42b838a50bb7fb23a331d596f7b.svg\" width=\"210\" height=\"26\" data-width=\"26.222\" data-height=\"3.297\" data-vertical-align=\"-1.083\" style=\"width: 26.222ex; height: 3.297ex; vertical-align: -1.083ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d4b\/d4b7e42b838a50bb7fb23a331d596f7b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d4b\/d4b7e42b838a50bb7fb23a331d596f7b.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0438 \u041b\u0435\u0436\u0430\u043d\u0434\u0440 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u043b \u043e\u0431\u043c\u0435\u043d \u0434\u0432\u0443\u0445 \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u044b\u0445 \u0432\u0435\u043b\u0438\u0447\u0438\u043d. \u0422\u043e\u0442 \u0444\u0430\u043a\u0442, \u0447\u0442\u043e \u0447\u0430\u0441\u0442\u043e \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432\u0430 \u043f\u0430\u0440\u0430 \u0441\u043e\u0441\u0442\u043e\u0438\u0442 \u0438\u0437 \u044d\u043a\u0441\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u043e\u0439 + \u0438\u043d\u0442\u0435\u043d\u0441\u0438\u0432\u043d\u043e\u0439 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b, \u044d\u0442\u043e \u043f\u0440\u043e\u0441\u0442\u043e \u0441\u043b\u0435\u0434\u0441\u0442\u0432\u0438\u0435 \u043e\u0434\u043d\u043e\u0440\u043e\u0434\u043d\u043e\u0441\u0442\u0438 \u0432\u0437\u044f\u0442\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439.<\/p>\n<p>\u041c\u044b \u0437\u0430\u0442\u0440\u043e\u043d\u0443\u043b\u0438 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u044b\u0439 \u043c\u043e\u043c\u0435\u043d\u0442 \u043e \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u043e\u0432. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u044d\u0442\u0430 \u0432\u0435\u0442\u043a\u0430 \u043e\u0442\u043a\u0440\u044b\u0432\u0430\u0435\u0442 \u0440\u044f\u0434 \u0446\u0435\u043d\u043d\u044b\u0445 \u0441\u0432\u044f\u0437\u0435\u0439<\/p>\n<p>\u042d\u043d\u0442\u0440\u043e\u043f\u0438\u044f<\/p>\n<p>\u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0441\u043c\u044b\u0441\u043b \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u043f\u043e\u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u0430\u043a. \u041f\u0443\u0441\u0442\u044c \u0432 \u044f\u0449\u0438\u043a\u0435 \u0441 \u0432\u044b\u043c\u044b\u0448\u043b\u0435\u043d\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u0433\u043e\u0440\u043e\u0434\u043a\u043e\u0439 \u043d\u0430\u0445\u043e\u0434\u0438\u0442\u0441\u044f \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0447\u0430\u0441\u0442\u0438\u0446. \u0421\u0443\u043c\u043c\u0430\u0440\u043d\u043e\u0435 \u043a\u043e\u043b-\u0432\u043e \u0447\u0430\u0441\u0442\u0438\u0446 \u2013 \u043c\u0430\u043a\u0440\u043e\u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u044f\u0449\u0438\u043a\u0430, \u0430 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0447\u0430\u0441\u0442\u0438\u0446 \u043f\u043e \u044f\u0447\u0435\u0439\u043a\u0430\u043c \u2013 \u043c\u0438\u043a\u0440\u043e\u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u044f. \u0412\u0441\u0435\u0433\u043e \u043c\u0438\u043a\u0440\u043e\u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0439, \u0440\u0435\u0430\u043b\u0438\u0437\u0443\u044e\u0449\u0438\u0445 \u0434\u0430\u043d\u043d\u043e\u0435 \u043c\u0430\u043a\u0440\u043e\u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0435, &#8212; <img decoding=\"async\" class=\"formula inline\" source=\"\\Omega\" alt=\"\\Omega\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2e\/2e9\/2e9ef3d6ef62a48d70720728d3e90e31.svg\" width=\"13\" height=\"18\" data-width=\"1.633\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.633ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2e\/2e9\/2e9ef3d6ef62a48d70720728d3e90e31.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2e\/2e9\/2e9ef3d6ef62a48d70720728d3e90e31.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0448\u0442\u0443\u043a, \u0432\u0441\u0435 \u043e\u043d\u0438 \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u044b \u0438 \u0440\u0430\u0432\u043d\u043e\u0432\u0435\u0440\u043e\u044f\u0442\u043d\u044b (\u0441 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c\u044e <img decoding=\"async\" class=\"formula inline\" source=\"p = \\frac{1}{\\Omega}\" alt=\"p = \\frac{1}{\\Omega}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2c\/2c7\/2c731e3f63a31537feab0b799e109c70.svg\" width=\"54\" height=\"37\" data-width=\"6.784\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 6.784ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2c\/2c7\/2c731e3f63a31537feab0b799e109c70.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2c\/2c7\/2c731e3f63a31537feab0b799e109c70.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u042d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0434\u043e\u043b\u0436\u043d\u0430 \u0431\u044b\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0435\u0439 \u043e\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\Omega\" alt=\"\\Omega\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2e\/2e9\/2e9ef3d6ef62a48d70720728d3e90e31.svg\" width=\"13\" height=\"18\" data-width=\"1.633\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.633ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2e\/2e9\/2e9ef3d6ef62a48d70720728d3e90e31.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2e\/2e9\/2e9ef3d6ef62a48d70720728d3e90e31.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u043f\u0440\u043e\u0441\u0442\u043e \u0431\u043e\u043b\u044c\u0448\u0435 \u043d\u0435 \u043e\u0442 \u0447\u0435\u0433\u043e \u043e\u0442\u0442\u043e\u043b\u043a\u043d\u0443\u0442\u044c\u0441\u044f. \u041f\u0440\u0438 \u044d\u0442\u043e\u043c \u043c\u044b \u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0430\u0434\u0434\u0438\u0442\u0438\u0432\u043d\u0430 (<img decoding=\"async\" class=\"formula inline\" source=\"S = S_{1} + S_{2}\" alt=\"S = S_{1} + S_{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdf\/fdfe448d271013e48c976933fcc00d49.svg\" width=\"96\" height=\"15\" data-width=\"11.991\" data-height=\"1.934\" data-vertical-align=\"-0.339\" style=\"width: 11.991ex; height: 1.934ex; vertical-align: -0.339ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdf\/fdfe448d271013e48c976933fcc00d49.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdf\/fdfe448d271013e48c976933fcc00d49.svg 781w\" loading=\"lazy\" decode=\"async\"\/>), \u0430 \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432 \u043c\u0438\u043a\u0440\u043e\u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0439 \u043c\u0443\u043b\u044c\u0442\u0438\u043f\u043b\u0438\u043a\u0430\u0442\u0438\u0432\u043d\u043e, \u043f\u043e \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0438 \u0441 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044f\u043c\u0438 \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u044b\u0445 \u0441\u043e\u0431\u044b\u0442\u0438\u0439 (<img decoding=\"async\" class=\"formula inline\" source=\"\\Omega = \\Omega_{1}\\Omega_{2}\" alt=\"\\Omega = \\Omega_{1}\\Omega_{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e09\/e09930995d38fac1f62216b71164c6de.svg\" width=\"79\" height=\"18\" data-width=\"9.893\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 9.893ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e09\/e09930995d38fac1f62216b71164c6de.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e09\/e09930995d38fac1f62216b71164c6de.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u0415\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u0438\u0442 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u0441\u0443\u043c\u043c\u0443 (<img decoding=\"async\" class=\"formula inline\" source=\"S\\left( \\Omega_{1}\\Omega_{2} \\right) = S\\left( \\Omega_{1} \\right) + S\\left( \\Omega_{2} \\right)\" alt=\"S\\left( \\Omega_{1}\\Omega_{2} \\right) = S\\left( \\Omega_{1} \\right) + S\\left( \\Omega_{2} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1c\/1c3\/1c359abe0119af87283acc9ece373769.svg\" width=\"216\" height=\"18\" data-width=\"27.057\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 27.057ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1c\/1c3\/1c359abe0119af87283acc9ece373769.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1c\/1c3\/1c359abe0119af87283acc9ece373769.svg 781w\" loading=\"lazy\" decode=\"async\"\/>) \u2013 \u044d\u0442\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c, \u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"S = k\\ln\\Omega\" alt=\"S = k\\ln\\Omega\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3e\/3ed\/3edeeb7f8a46efd6c1f29119ebabc84d.svg\" width=\"79\" height=\"18\" data-width=\"9.93\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 9.93ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3e\/3ed\/3edeeb7f8a46efd6c1f29119ebabc84d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3e\/3ed\/3edeeb7f8a46efd6c1f29119ebabc84d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0432 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0432 \u0437\u0430\u043a\u043e\u043d \u0438\u0434\u0435\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0433\u0430\u0437\u0430, \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043c, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"k\" alt=\"k\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg\" width=\"12\" height=\"13\" data-width=\"1.179\" data-height=\"1.595\" data-vertical-align=\"-0.025\" style=\"width: 1.179ex; height: 1.595ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 \u0411\u043e\u043b\u044c\u0446\u043c\u0430\u043d\u0430. \u0421\u0430\u043c \u0436\u0435 \u044f\u0449\u0438\u043a \u043d\u0435 \u043e\u0431\u044f\u0437\u0430\u043d \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0442\u044c \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u0447\u0430\u0441\u0442\u0438\u0446: \u043e\u043d \u043c\u043e\u0436\u0435\u0442, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0438\u043b\u043b\u044e\u0441\u0442\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0447\u0430\u0441\u0442\u0438\u0446 \u043f\u043e \u044d\u043d\u0435\u0440\u0433\u0438\u044f\u043c. \u0412 \u0438\u0442\u043e\u0433\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"S = k\\ln{\\Omega(E)}\" alt=\"S = k\\ln{\\Omega(E)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/bef\/bef239aafe934d4f95256dbb856f2dc6.svg\" width=\"107\" height=\"18\" data-width=\"13.418\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.418ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/bef\/bef239aafe934d4f95256dbb856f2dc6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/bef\/bef239aafe934d4f95256dbb856f2dc6.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u044b\u0440\u0430\u0437\u0438\u043c \u043e\u0431\u0449\u0443\u044e \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u0447\u0435\u0440\u0435\u0437 \u0440\u0430\u0432\u043d\u044b\u0435 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"p(E) = \\frac{1}{\\Omega(E)}\" alt=\"p(E) = \\frac{1}{\\Omega(E)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e7\/e72\/e7294b169bc59a742679933e04870286.svg\" width=\"110\" height=\"42\" data-width=\"13.761\" data-height=\"5.208\" data-vertical-align=\"-2.038\" style=\"width: 13.761ex; height: 5.208ex; vertical-align: -2.038ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e7\/e72\/e7294b169bc59a742679933e04870286.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e7\/e72\/e7294b169bc59a742679933e04870286.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"S = k\\ln{\\Omega(E)} = k \\cdot \\left( - \\frac{\\Omega}{\\Omega} \\right) \\cdot \\left( - \\ln\\Omega \\right) = - k\\sum_{i = 1}^{\\Omega}{\\frac{1}{\\Omega}\\ln\\frac{1}{\\Omega}} = - k\\sum_{i = 1}^{\\Omega}{p\\ln p}\" alt=\"S = k\\ln{\\Omega(E)} = k \\cdot \\left( - \\frac{\\Omega}{\\Omega} \\right) \\cdot \\left( - \\ln\\Omega \\right) = - k\\sum_{i = 1}^{\\Omega}{\\frac{1}{\\Omega}\\ln\\frac{1}{\\Omega}} = - k\\sum_{i = 1}^{\\Omega}{p\\ln p}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e05\/e05f6e42ef392cefda781689a3ed9d4d.svg\" width=\"550\" height=\"54\" data-width=\"68.791\" data-height=\"6.773\" data-vertical-align=\"-2.821\" style=\"width: 68.791ex; height: 6.773ex; vertical-align: -2.821ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e05\/e05f6e42ef392cefda781689a3ed9d4d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e05\/e05f6e42ef392cefda781689a3ed9d4d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e \u043f\u043e\u0445\u043e\u0436\u0435 \u043d\u0430 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u0428\u0435\u043d\u043d\u043e\u043d\u0430 (\u043e\u0431 \u044d\u0442\u043e\u043c \u2013 \u043f\u043e\u0437\u0436\u0435)<\/p>\n<p>\u0412\u0442\u043e\u0440\u043e\u0435 \u043d\u0430\u0447\u0430\u043b\u043e \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u043a\u0438 \u0433\u043b\u0430\u0441\u0438\u0442, \u0447\u0442\u043e \u0432 \u0438\u0437\u043e\u043b\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u0439 \u0441\u0438\u0441\u0442\u0435\u043c\u0435 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0440\u0430\u0441\u0442\u0435\u0442, \u043f\u043e\u043a\u0430 \u043d\u0435 \u0434\u043e\u0441\u0442\u0438\u0433\u043d\u0435\u0442 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430 \u0432 \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u043c \u0440\u0430\u0432\u043d\u043e\u0432\u0435\u0441\u0438\u0438. \u041f\u043e \u0432\u0438\u0434\u0443 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u044f\u0441\u043d\u043e, \u0447\u0442\u043e \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u0430 \u043d\u0430 \u0440\u0430\u0432\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0438. \u0427\u0430\u0441\u0442\u043e 2\u0435 \u043d\u0430\u0447\u0430\u043b\u043e \u0438\u0441\u043a\u0430\u0436\u0435\u043d\u043d\u043e \u043f\u0435\u0440\u0435\u0441\u043a\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u043a\u0430\u043a \u00ab\u0432\u0441\u0435\u043b\u0435\u043d\u043d\u0430\u044f \u0441\u0442\u0440\u0435\u043c\u0438\u0442\u0441\u044f \u043a \u0431\u0435\u0441\u043f\u043e\u0440\u044f\u0434\u043a\u0443\u00bb, \u0445\u043e\u0442\u044f \u0442\u043e\u0447\u043d\u0435\u0435 \u0431\u044b\u043b\u043e \u0431\u044b \u00ab\u0432\u0441\u0435\u043b\u0435\u043d\u043d\u0443\u044e \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u0435\u0435 \u0437\u0430\u0441\u0442\u0430\u0442\u044c \u0432 \u043d\u0430\u0438\u0431\u043e\u043b\u0435\u0435 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u043c \u0435\u0435 \u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0438, \u0438 \u0442\u0430\u043a\u0438\u043c \u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0435\u043c \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u043e, \u0447\u0442\u043e \u0447\u0430\u0441\u0442\u043e \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u0435\u0442 \u0440\u0430\u0432\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u043c\u0443 \u0440\u0430\u0437\u0431\u0440\u043e\u0441\u0443\u00bb. \u041d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435, \u0440\u043e\u0441\u0442 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u2013 \u044d\u0442\u043e \u043f\u0440\u043e\u0441\u0442\u043e \u0441\u043b\u0435\u0434\u0441\u0442\u0432\u0438\u0435 \u0442\u043e\u0433\u043e \u0444\u0430\u043a\u0442\u0430, \u0447\u0442\u043e \u0440\u0430\u0432\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u0435\u0442 \u043d\u0430\u0438\u0431\u043e\u043b\u044c\u0448\u0435\u043c\u0443 \u0447\u0438\u0441\u043b\u0443 \u043c\u0438\u043a\u0440\u043e\u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0439 \u0441\u0440\u0435\u0434\u0438 \u0432\u0441\u0435\u0445 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0439. \u042d\u0442\u0430 \u0438\u0434\u0435\u044f \u043b\u0435\u0436\u0438\u0442 \u0432 \u043e\u0441\u043d\u043e\u0432\u0435 \u043f\u0440\u0438\u043d\u0446\u0438\u043f\u0430 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438<\/p>\n<p>\u0421\u0432\u043e\u0431\u043e\u0434\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f<\/p>\n<p>\u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043d\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"p_{i}\" alt=\"p_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8a\/8a4\/8a4bbd153c74655abb7ca04c0fa901d8.svg\" width=\"15\" height=\"12\" data-width=\"1.878\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.878ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8a\/8a4\/8a4bbd153c74655abb7ca04c0fa901d8.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8a\/8a4\/8a4bbd153c74655abb7ca04c0fa901d8.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043c\u0438\u043a\u0440\u043e\u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"i\" alt=\"i\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/865\/865c0c0b4ab0e063e5caa3387c1a8741.svg\" width=\"12\" height=\"12\" data-width=\"0.781\" data-height=\"1.52\" data-vertical-align=\"-0.025\" style=\"width: 0.781ex; height: 1.52ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/865\/865c0c0b4ab0e063e5caa3387c1a8741.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/865\/865c0c0b4ab0e063e5caa3387c1a8741.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441 \u044d\u043d\u0435\u0440\u0433\u0438\u0435\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"E_{i}\" alt=\"E_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/82d\/82deb5e05e05a48f33fea4dd071e001d.svg\" width=\"19\" height=\"15\" data-width=\"2.409\" data-height=\"1.895\" data-vertical-align=\"-0.357\" style=\"width: 2.409ex; height: 1.895ex; vertical-align: -0.357ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/82d\/82deb5e05e05a48f33fea4dd071e001d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/82\/82d\/82deb5e05e05a48f33fea4dd071e001d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432 \u0441\u0438\u0441\u0442\u0435\u043c\u0435 \u0438\u0437 \u044f\u0449\u0438\u043a\u0430 \u0438 \u0440\u0435\u0437\u0435\u0440\u0432\u0443\u0430\u0440\u0430. \u0412\u0441\u0435 \u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u0440\u0430\u0432\u043d\u043e\u0432\u0435\u0440\u043e\u044f\u0442\u043d\u044b, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c \u043f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u0430 \u0447\u0438\u0441\u043b\u0443 \u0441\u043f\u043e\u0441\u043e\u0431\u043e\u0432 \u0435\u0433\u043e \u0440\u0435\u0430\u043b\u0438\u0437\u043e\u0432\u0430\u0442\u044c: <img decoding=\"async\" class=\"formula inline\" source=\"p_{i}\\ \\sim\\ \\Omega\\left( E - E_{i} \\right)\" alt=\"p_{i}\\ \\sim\\ \\Omega\\left( E - E_{i} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/97\/97d\/97d24c3655eb33d231417265eb6f203c.svg\" width=\"134\" height=\"18\" data-width=\"16.7\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 16.7ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/97\/97d\/97d24c3655eb33d231417265eb6f203c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/97\/97d\/97d24c3655eb33d231417265eb6f203c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u0440\u0438\u0447\u0435\u043c \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u044f\u0449\u0438\u043a\u0430 \u043c\u043d\u043e\u0433\u043e \u043c\u0435\u043d\u044c\u0448\u0435 \u044d\u043d\u0435\u0440\u0433\u0438\u0438 \u0440\u0435\u0437\u0435\u0440\u0432\u0443\u0430\u0440\u0430 (<img decoding=\"async\" class=\"formula inline\" source=\"E \\gg E_{i}\" alt=\"E \\gg E_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8d\/8de\/8ded183dae10aa6ae36773ddbc552b9e.svg\" width=\"61\" height=\"18\" data-width=\"7.657\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.657ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8d\/8de\/8ded183dae10aa6ae36773ddbc552b9e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8d\/8de\/8ded183dae10aa6ae36773ddbc552b9e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u0420\u0430\u0437\u043b\u043e\u0436\u0438\u0432 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u043f\u043e \u0422\u0435\u0439\u043b\u043e\u0440\u0443, \u043f\u043e\u043b\u0443\u0447\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{S}{k} = \\ln{\\Omega\\left( E - E_{i} \\right)} \\approx \\ln{\\Omega(E)} - \\frac{\\partial\\ln\\Omega}{\\partial E}E_{i}\" alt=\"\\frac{S}{k} = \\ln{\\Omega\\left( E - E_{i} \\right)} \\approx \\ln{\\Omega(E)} - \\frac{\\partial\\ln\\Omega}{\\partial E}E_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b85\/b85dd8c77b6e146e1090edc77e138aa4.svg\" width=\"324\" height=\"38\" data-width=\"40.539\" data-height=\"4.749\" data-vertical-align=\"-1.809\" style=\"width: 40.539ex; height: 4.749ex; vertical-align: -1.809ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b85\/b85dd8c77b6e146e1090edc77e138aa4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b85\/b85dd8c77b6e146e1090edc77e138aa4.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u044b, <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\partial S}{\\partial E} = \\frac{1}{T}\" alt=\"\\frac{\\partial S}{\\partial E} = \\frac{1}{T}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/85\/85b\/85b192d1cecd7253b454a1949527fe9c.svg\" width=\"77\" height=\"38\" data-width=\"9.61\" data-height=\"4.749\" data-vertical-align=\"-1.809\" style=\"width: 9.61ex; height: 4.749ex; vertical-align: -1.809ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/85\/85b\/85b192d1cecd7253b454a1949527fe9c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/85\/85b\/85b192d1cecd7253b454a1949527fe9c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{S}{k} \\approx \\ln{\\Omega(E)} - \\frac{E_{i}}{kT}\" alt=\"\\frac{S}{k} \\approx \\ln{\\Omega(E)} - \\frac{E_{i}}{kT}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/67\/674\/674daa2ca54a20e32b804ed60e811c3c.svg\" width=\"155\" height=\"38\" data-width=\"19.391\" data-height=\"4.701\" data-vertical-align=\"-1.785\" style=\"width: 19.391ex; height: 4.701ex; vertical-align: -1.785ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/67\/674\/674daa2ca54a20e32b804ed60e811c3c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/67\/674\/674daa2ca54a20e32b804ed60e811c3c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0422\u043e\u0433\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\ln p_{i}\\sim\\ln{\\Omega(E)} - \\frac{E_{i}}{kT}\" alt=\"\\ln p_{i}\\sim\\ln{\\Omega(E)} - \\frac{E_{i}}{kT}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e01\/e01ab079080db0e9fb17b6bce3026cda.svg\" width=\"169\" height=\"37\" data-width=\"21.078\" data-height=\"4.645\" data-vertical-align=\"-1.757\" style=\"width: 21.078ex; height: 4.645ex; vertical-align: -1.757ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e01\/e01ab079080db0e9fb17b6bce3026cda.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e01\/e01ab079080db0e9fb17b6bce3026cda.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412\u0437\u044f\u0432 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u0443 \u0438 \u0443\u0447\u0442\u044f \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0443, \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0413\u0438\u0431\u0431\u0441\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p_{i} = \\frac{1}{Z}e^{- E_{i}\/kT} = \\frac{e^{- E_{i}\/kT}}{\\sum_{i}^{}e^{- E_{i}\/kT}}\" alt=\"p_{i} = \\frac{1}{Z}e^{- E_{i}\/kT} = \\frac{e^{- E_{i}\/kT}}{\\sum_{i}^{}e^{- E_{i}\/kT}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a27\/a2761087b0948d6ee50a7ef20314ea23.svg\" width=\"232\" height=\"48\" data-width=\"28.945\" data-height=\"5.977\" data-vertical-align=\"-2.423\" style=\"width: 28.945ex; height: 5.977ex; vertical-align: -2.423ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a27\/a2761087b0948d6ee50a7ef20314ea23.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a27\/a2761087b0948d6ee50a7ef20314ea23.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e \u043f\u043e\u0445\u043e\u0436\u0435 \u043d\u0430 \u0441\u043e\u0444\u0442\u043c\u0430\u043a\u0441 \u0441 \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u043e\u0439 (\u043e\u0431 \u044d\u0442\u043e\u043c \u2013 \u043f\u043e\u0437\u0436\u0435)<\/p>\n<p>\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u043c \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 \u0432 \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 (\u0432\u0441\u043f\u043e\u043c\u043d\u0438\u0432 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0432\u043d\u0443\u0442\u0440\u0435\u043d\u043d\u0435\u0439 \u044d\u043d\u0435\u0440\u0433\u0438\u0438):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"S = - k\\sum_{}^{}{p_{i}\\ln p_{i}} = - k\\sum_{}^{}{p_{i}\\left( - \\ln Z - \\frac{E_{i}}{kT} \\right)} = k\\ln Z + \\frac{U}{T}\\ \\\u00a0 \\Rightarrow \\ \\ U - TS = - kT\\ln Z\" alt=\"S = - k\\sum_{}^{}{p_{i}\\ln p_{i}} = - k\\sum_{}^{}{p_{i}\\left( - \\ln Z - \\frac{E_{i}}{kT} \\right)} = k\\ln Z + \\frac{U}{T}\\ \\\u00a0 \\Rightarrow \\ \\ U - TS = - kT\\ln Z\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b4\/6b4f31b3e8490a96e51b2f4fd484c709.svg\" width=\"681\" height=\"47\" data-width=\"85.094\" data-height=\"5.881\" data-vertical-align=\"-2.375\" style=\"width: 85.094ex; height: 5.881ex; vertical-align: -2.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b4\/6b4f31b3e8490a96e51b2f4fd484c709.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b4\/6b4f31b3e8490a96e51b2f4fd484c709.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0410 \u043e\u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"U - TS\" alt=\"U - TS\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/08\/081\/081dd3a00b89cf708413a016c1bd40d2.svg\" width=\"60\" height=\"14\" data-width=\"7.553\" data-height=\"1.781\" data-vertical-align=\"-0.186\" style=\"width: 7.553ex; height: 1.781ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/08\/081\/081dd3a00b89cf708413a016c1bd40d2.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/08\/081\/081dd3a00b89cf708413a016c1bd40d2.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u043e\u0439 \u044d\u043d\u0435\u0440\u0433\u0438\u0438, \u043f\u043e\u044d\u0442\u043e\u043c\u0443:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"F = - \\frac{\\ln Z}{\\beta},\\ \\ \u0433\u0434\u0435\\ Z = \\sum_{i}^{}e^{- \\beta E_{i}}\\ \u0438\\ \\ \\beta = \\frac{1}{kT}\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ F = - kT\\ln\\left\\lbrack \\sum_{i}^{}e^{- E_{i}\/kT} \\right\\rbrack\" alt=\"F = - \\frac{\\ln Z}{\\beta},\\ \\ \u0433\u0434\u0435\\ Z = \\sum_{i}^{}e^{- \\beta E_{i}}\\ \u0438\\ \\ \\beta = \\frac{1}{kT}\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ F = - kT\\ln\\left\\lbrack \\sum_{i}^{}e^{- E_{i}\/kT} \\right\\rbrack\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/80\/804\/804a3f26c00f53beeb6b45d43b7be121.svg\" width=\"597\" height=\"54\" data-width=\"74.683\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 74.683ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/80\/804\/804a3f26c00f53beeb6b45d43b7be121.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/80\/804\/804a3f26c00f53beeb6b45d43b7be121.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e \u043f\u043e\u0445\u043e\u0436\u0435 \u043d\u0430 log-sum-exp (\u043e\u0431 \u044d\u0442\u043e\u043c \u2013 \u043f\u043e\u0437\u0436\u0435)<\/p>\n<p>\u0421\u0432\u044f\u0437\u044c \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u0438 \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u043e\u0439 \u044d\u043d\u0435\u0440\u0433\u0438\u0438<\/p>\n<p>\u041f\u0440\u043e\u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0443\u0435\u043c \u043f\u043e \u0443\u0440\u043e\u0432\u043d\u044f\u043c \u044d\u043d\u0435\u0440\u0433\u0438\u0438 \u0447\u0435\u0440\u0435\u0437 \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c \u0441\u043e\u0441\u0442\u043e\u044f\u043d\u0438\u0439:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"Z(\\beta) = \\int_{}^{}{\\Omega(E)e^{- \\beta E}dE} = \\int_{}^{}e^{S(E)\/k\\\u00a0 - \\beta E}dE,\\ \\ \u0433\u0434\u0435\\ S,E\\ \\sim\\ N\" alt=\"Z(\\beta) = \\int_{}^{}{\\Omega(E)e^{- \\beta E}dE} = \\int_{}^{}e^{S(E)\/k\\\u00a0 - \\beta E}dE,\\ \\ \u0433\u0434\u0435\\ S,E\\ \\sim\\ N\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2a\/2a1\/2a185a067b0670ac1c3d6f4ef2d3b9b7.svg\" width=\"474\" height=\"41\" data-width=\"59.245\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 59.245ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2a\/2a1\/2a185a067b0670ac1c3d6f4ef2d3b9b7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2a\/2a1\/2a185a067b0670ac1c3d6f4ef2d3b9b7.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u0438 \u043e\u0433\u0440\u043e\u043c\u043d\u043e\u043c \u0447\u0438\u0441\u043b\u0435 \u0447\u0430\u0441\u0442\u0438\u0446 (<img decoding=\"async\" class=\"formula inline\" source=\"N \\rightarrow \\infty\" alt=\"N \\rightarrow \\infty\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/52\/524\/524ce044c402a61b400bc14273b8cdda.svg\" width=\"62\" height=\"18\" data-width=\"7.791\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.791ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/52\/524\/524ce044c402a61b400bc14273b8cdda.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/52\/524\/524ce044c402a61b400bc14273b8cdda.svg 781w\" loading=\"lazy\" decode=\"async\"\/>) \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0426\u041f\u0422. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u043e\u0434\u044b\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u044c\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u043e\u043d\u0446\u0435\u043d\u0442\u0440\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u0432\u0431\u043b\u0438\u0437\u0438 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430, \u0432\u044b\u0440\u043e\u0436\u0434\u0430\u0435\u0442\u0441\u044f \u0432 \u0434\u0435\u043b\u044c\u0442\u0430-\u0444\u0443\u043d\u043a\u0446\u0438\u044e, \u0438 \u0432\u043a\u043b\u0430\u0434 \u0432 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b \u0434\u0430\u0435\u0442 \u043b\u0438\u0448\u044c \u043e\u0434\u043d\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043f\u043e\u0434\u044b\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u044c\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u2013 \u0432 \u0435\u0435 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0435 (\u044d\u0442\u043e \u043d\u0430 \u043f\u0430\u043b\u044c\u0446\u0430\u0445 \u043c\u0435\u0442\u043e\u0434 \u043f\u0435\u0440\u0435\u0432\u0430\u043b\u0430):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"Z(\\beta) = \\exp\\left\\lbrack \\sup_{E}\\left( \\frac{S(E)}{k}\\\u00a0 - \\beta E \\right) \\right\\rbrack\" alt=\"Z(\\beta) = \\exp\\left\\lbrack \\sup_{E}\\left( \\frac{S(E)}{k}\\\u00a0 - \\beta E \\right) \\right\\rbrack\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a14\/a146f5a4f87a0c31f5b95b743aa9e3f9.svg\" width=\"267\" height=\"44\" data-width=\"33.315\" data-height=\"5.451\" data-vertical-align=\"-2.16\" style=\"width: 33.315ex; height: 5.451ex; vertical-align: -2.16ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a14\/a146f5a4f87a0c31f5b95b743aa9e3f9.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a14\/a146f5a4f87a0c31f5b95b743aa9e3f9.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u043e\u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u043e\u0432\u0430\u0432 <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\partial}{\\partial E}\\left( \\frac{S(E)}{k}\\\u00a0 - \\beta E \\right) = 0\" alt=\"\\frac{\\partial}{\\partial E}\\left( \\frac{S(E)}{k}\\\u00a0 - \\beta E \\right) = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/906\/9061618b5130022cca66f81136919b5e.svg\" width=\"190\" height=\"44\" data-width=\"23.767\" data-height=\"5.451\" data-vertical-align=\"-2.16\" style=\"width: 23.767ex; height: 5.451ex; vertical-align: -2.16ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/906\/9061618b5130022cca66f81136919b5e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/906\/9061618b5130022cca66f81136919b5e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u044b: <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\partial S}{\\partial E} = \\frac{1}{T}\" alt=\"\\frac{\\partial S}{\\partial E} = \\frac{1}{T}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/85\/85b\/85b192d1cecd7253b454a1949527fe9c.svg\" width=\"77\" height=\"38\" data-width=\"9.61\" data-height=\"4.749\" data-vertical-align=\"-1.809\" style=\"width: 9.61ex; height: 4.749ex; vertical-align: -1.809ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/85\/85b\/85b192d1cecd7253b454a1949527fe9c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/85\/85b\/85b192d1cecd7253b454a1949527fe9c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041a\u0430\u043a \u0443\u0436\u0435 \u0431\u044b\u043b\u043e \u0432 \u0440\u0430\u0437\u0434\u0435\u043b\u0435 \u043f\u0440\u043e \u041b\u0430\u0433\u0440\u00e0\u043d\u0436\u0430 \u0438 \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0430 \u0441 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c \u0438\u043c\u043f\u0443\u043b\u044c\u0441\u0430, \u0442\u0443\u0442 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u044b \u0442\u043e\u0436\u0435 \u0432\u043e\u0437\u043d\u0438\u043a\u043b\u043e \u0438\u0437 \u043f\u043e\u0438\u0441\u043a\u0430 \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0443\u043c\u0430. \u041c\u044b \u043d\u0430\u0448\u043b\u0438 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"S(E)\" alt=\"S(E)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5f\/5fc\/5fc3255378cfa40548e20d97f642cfa1.svg\" width=\"40\" height=\"18\" data-width=\"4.948\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.948ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5f\/5fc\/5fc3255378cfa40548e20d97f642cfa1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5f\/5fc\/5fc3255378cfa40548e20d97f642cfa1.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0432\u043e\u0433\u043d\u0443\u0442\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f (<img decoding=\"async\" class=\"formula inline\" source=\"- S\" alt=\"- S\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/365\/36551ab0f34d71f17df6aa6e42d018eb.svg\" width=\"26\" height=\"14\" data-width=\"3.219\" data-height=\"1.781\" data-vertical-align=\"-0.186\" style=\"width: 3.219ex; height: 1.781ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/365\/36551ab0f34d71f17df6aa6e42d018eb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/365\/36551ab0f34d71f17df6aa6e42d018eb.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f), \u0440\u0435\u0430\u043b\u0438\u0437\u043e\u0432\u0430\u0432 \u043f\u0440\u0438\u043d\u0446\u0438\u043f \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438. \u042d\u0442\u043e \u043b\u043e\u0433\u0438\u0447\u043d\u043e: \u043d\u0430\u043b\u0438\u0447\u0438\u0435 \u0443\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0432\u0448\u0435\u0439\u0441\u044f \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u044b \u2013 \u043f\u043e\u043d\u044f\u0442\u043d\u044b\u0439 \u043f\u0440\u0438\u0437\u043d\u0430\u043a \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0440\u0430\u0432\u043d\u043e\u0432\u0435\u0441\u0438\u044f, \u0433\u0434\u0435 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u0430.<\/p>\n<p>\u0412\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043a \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u043e\u0439 \u044d\u043d\u0435\u0440\u0433\u0438\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"- \\beta F(\\beta) = \\ln Z = \\sup_{E}\\left( \\frac{S(E)}{k}\\\u00a0 - \\beta E \\right) = \\sup_{E}\\left( - \\beta \\cdot E\\\u00a0 - \\left( - \\frac{S(E)}{k} \\right) \\right) = \\sup_{E}\\left( p \\cdot E - f(E) \\right)\" alt=\"- \\beta F(\\beta) = \\ln Z = \\sup_{E}\\left( \\frac{S(E)}{k}\\\u00a0 - \\beta E \\right) = \\sup_{E}\\left( - \\beta \\cdot E\\\u00a0 - \\left( - \\frac{S(E)}{k} \\right) \\right) = \\sup_{E}\\left( p \\cdot E - f(E) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/38\/384\/384b4f2fd6494e46a55c2f163761a4ca.svg\" width=\"709\" height=\"44\" data-width=\"88.57\" data-height=\"5.451\" data-vertical-align=\"-2.16\" style=\"width: 88.57ex; height: 5.451ex; vertical-align: -2.16ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/38\/384\/384b4f2fd6494e46a55c2f163761a4ca.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/38\/384\/384b4f2fd6494e46a55c2f163761a4ca.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c, \u043a\u043e\u0433\u0434\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"f(E) = - \\frac{S(E)}{k}\" alt=\"f(E) = - \\frac{S(E)}{k}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7a\/7af\/7afaa91fc86a87fc0c0e80320b9e5007.svg\" width=\"124\" height=\"39\" data-width=\"15.454\" data-height=\"4.88\" data-vertical-align=\"-1.874\" style=\"width: 15.454ex; height: 4.88ex; vertical-align: -1.874ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7a\/7af\/7afaa91fc86a87fc0c0e80320b9e5007.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7a\/7af\/7afaa91fc86a87fc0c0e80320b9e5007.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441\u0442\u0430\u043b\u0430 \u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f, \u043f\u0440\u043e\u044f\u0432\u0438\u043b\u043e\u0441\u044c \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 (\u0447\u0435\u0440\u0435\u0437 sup). \u0421\u0430\u043c\u0430 \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432\u0430 \u043f\u0430\u0440\u0430 \u2013 \u044d\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\ln Z\" alt=\"\\ln Z\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2f\/2f0\/2f0cb9ef056df530d020345ce3cb0d3d.svg\" width=\"31\" height=\"18\" data-width=\"3.9\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.9ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2f\/2f0\/2f0cb9ef056df530d020345ce3cb0d3d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2f\/2f0\/2f0cb9ef056df530d020345ce3cb0d3d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"- \\beta\" alt=\"- \\beta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c2\/3c2d85705ee2d88791be865b34c975a1.svg\" width=\"24\" height=\"18\" data-width=\"3.041\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.041ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c2\/3c2d85705ee2d88791be865b34c975a1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c2\/3c2d85705ee2d88791be865b34c975a1.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"- \\frac{S}{k}\" alt=\"- \\frac{S}{k}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d7\/d77\/d77257e1db78fa8af4c4a7ca87d17d39.svg\" width=\"34\" height=\"38\" data-width=\"4.215\" data-height=\"4.701\" data-vertical-align=\"-1.785\" style=\"width: 4.215ex; height: 4.701ex; vertical-align: -1.785ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d7\/d77\/d77257e1db78fa8af4c4a7ca87d17d39.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d7\/d77\/d77257e1db78fa8af4c4a7ca87d17d39.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"E\" alt=\"E\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a3\/3a3ea00cfc35332cedf6e5e9a32e94da.svg\" width=\"14\" height=\"12\" data-width=\"1.729\" data-height=\"1.538\" style=\"width: 1.729ex; height: 1.538ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a3\/3a3ea00cfc35332cedf6e5e9a32e94da.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a3\/3a3ea00cfc35332cedf6e5e9a32e94da.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0422\u0430\u043a\u0430\u044f \u043f\u0430\u0440\u0430 \u043c\u043e\u0436\u0435\u0442 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u044c\u0441\u044f \u0433\u0440\u043e\u043c\u043e\u0437\u0434\u043a\u043e\u0439, \u043d\u043e \u043d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u043e\u043d\u0430 \u0441\u043a\u0440\u044b\u0432\u0430\u0435\u0442 \u043a\u0440\u0430\u0441\u0438\u0432\u0443\u044e \u0441\u0432\u044f\u0437\u044c: \u043c\u044b \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u043b\u0438, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\ln Z\" alt=\"\\ln Z\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2f\/2f0\/2f0cb9ef056df530d020345ce3cb0d3d.svg\" width=\"31\" height=\"18\" data-width=\"3.9\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.9ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2f\/2f0\/2f0cb9ef056df530d020345ce3cb0d3d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2f\/2f0\/2f0cb9ef056df530d020345ce3cb0d3d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e log-sum-exp, \u0430 <img decoding=\"async\" class=\"formula inline\" source=\"- \\frac{S(E)}{k}\" alt=\"- \\frac{S(E)}{k}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/642\/64217252512045d3037a8f867ded6b63.svg\" width=\"62\" height=\"39\" data-width=\"7.704\" data-height=\"4.88\" data-vertical-align=\"-1.874\" style=\"width: 7.704ex; height: 4.88ex; vertical-align: -1.874ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/642\/64217252512045d3037a8f867ded6b63.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/64\/642\/64217252512045d3037a8f867ded6b63.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"- \\sum_{}^{}{p\\ln p}\" alt=\"- \\sum_{}^{}{p\\ln p}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/209\/209955a3b3ccefe07a46843cf77306bc.svg\" width=\"86\" height=\"43\" data-width=\"10.698\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 10.698ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/209\/209955a3b3ccefe07a46843cf77306bc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/209\/209955a3b3ccefe07a46843cf77306bc.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0418\u0442\u043e\u0433: log-sum-exp \u0438 \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432\u043e \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u044b<\/p>\n<p>\u0418\u0442\u043e\u0433: \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0432 \u0441\u0442\u0430\u0442\u0444\u0438\u0437\u0435 \u043f\u043e\u0440\u043e\u0436\u0434\u0430\u0435\u0442 \u043c\u043d\u043e\u0433\u043e \u043f\u043e\u043b\u0435\u0437\u043d\u044b\u0445 \u0434\u043b\u044f \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u0438 ML \u0438\u0434\u0435\u0439<\/p>\n<hr\/>\n<h2>\u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0430<\/h2>\n<p>\u0415\u0441\u043b\u0438 \u0437\u0430\u0434\u0430\u0447\u0430 \u0442\u0435\u0440\u0432\u0435\u0440\u0430 \u043f\u043e\u043d\u044f\u0442\u044c \u043f\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u043c \u043c\u043e\u0434\u0435\u043b\u0438, \u043a\u0430\u043a\u0438\u0435 \u0434\u0430\u043d\u043d\u044b\u0435 \u043e\u043d\u0430 \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0443\u0435\u0442, \u0442\u043e \u0437\u0430\u0434\u0430\u0447\u0430 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u2013 \u043f\u043e \u0434\u0430\u043d\u043d\u044b\u043c \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u043e\u0446\u0435\u043d\u043a\u0443 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u043e\u0432. \u0417\u0430\u043c\u0435\u0442\u043d\u0430\u044f \u0447\u0430\u0441\u0442\u044c \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0430 \u043d\u0430 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0438 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430<\/p>\n<p>\u042d\u043d\u0442\u0440\u043e\u043f\u0438\u044f, softmax, logsumexp<\/p>\n<p>\u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0440\u0430\u0437\u0434\u0435\u043b \u043f\u0440\u043e \u0441\u0442\u0430\u0442\u0444\u0438\u0437 (\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f, \u0413\u0438\u0431\u0431\u0441, \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f). \u0422\u0435\u043c, \u043a\u0442\u043e \u0441\u0442\u0430\u043b\u043a\u0438\u0432\u0430\u043b\u0441\u044f \u0441\u043e \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u043e\u0439\/ML, \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0432 \u0440\u0430\u0437\u0434\u0435\u043b\u0435 \u043f\u0440\u043e \u0441\u0442\u0430\u0442\u0444\u0438\u0437 \u0443\u0436\u0435 \u0437\u0430\u0441\u043f\u043e\u0439\u043b\u0435\u0440\u0438\u043b\u0438 3 \u0432\u0435\u0449\u0438:<\/p>\n<ol>\n<li>\n<p>\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0438\u043c\u0435\u0435\u0442 \u0432\u0438\u0434 <img decoding=\"async\" class=\"formula inline\" source=\"- \\sum_{}^{}{p\\ln p}\" alt=\"- \\sum_{}^{}{p\\ln p}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/209\/209955a3b3ccefe07a46843cf77306bc.svg\" width=\"86\" height=\"43\" data-width=\"10.698\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 10.698ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/209\/209955a3b3ccefe07a46843cf77306bc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/209\/209955a3b3ccefe07a46843cf77306bc.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043a\u0430\u043a \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0428\u0435\u043d\u043d\u043e\u043d\u0430<\/p>\n<\/li>\n<li>\n<p>\u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0433\u0438\u0431\u0431\u0441\u0430 \u043f\u043e\u0445\u043e\u0436\u0435 \u043d\u0430 softmax \u0441 \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u043e\u0439<\/p>\n<\/li>\n<li>\n<p>\u0441\u0432\u043e\u0431\u043e\u0434\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u0438\u043c\u0435\u0435\u0442 \u0432\u0438\u0434 log-sum-exp<\/p>\n<\/li>\n<\/ol>\n<p>\u0412\u043e\u0437\u044c\u043c\u0435\u043c \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u0428\u0435\u043d\u043d\u043e\u043d\u0430 \u043d\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u043d\u043e\u043c \u0441\u0438\u043c\u043f\u043b\u0435\u043a\u0441\u0435. \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u044b\u0435 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 (\u0432 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u044b, \u043d\u043e \u0441 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0430\u043c\u0438 \u0432\u043c\u0435\u0441\u0442\u043e \u0441\u0443\u043c\u043c). \u041f\u043b\u044e\u0441 \u0434\u043e\u0433\u043e\u0432\u043e\u0440\u0438\u043c\u0441\u044f \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0432\u043c\u0435\u0441\u0442\u043e \u0434\u0432\u043e\u0438\u0447\u043d\u043e\u0433\u043e:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"- H(p) = \\sum_{i}^{}{p_{i}\\ln p_{i}},\\ \\ p \\in \\Delta = \\left\\{ p:\\ \\ \\ p_{i} \\geq 0\\ \\ ,\\ \\sum_{i}^{}p_{i} = 1 \\right\\}\" alt=\"- H(p) = \\sum_{i}^{}{p_{i}\\ln p_{i}},\\ \\ p \\in \\Delta = \\left\\{ p:\\ \\ \\ p_{i} \\geq 0\\ \\ ,\\ \\sum_{i}^{}p_{i} = 1 \\right\\}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3b\/3b8\/3b841ea1e0032c0b09ed67c0db91b336.svg\" width=\"462\" height=\"54\" data-width=\"57.812\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 57.812ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3b\/3b8\/3b841ea1e0032c0b09ed67c0db91b336.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3b\/3b8\/3b841ea1e0032c0b09ed67c0db91b336.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0435\u0435 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\nabla\\left( - H(p) \\right) = \\begin{bmatrix} \\ln p_{1} + 1 &amp; \\ldots &amp; \\ln p_{n} + 1 \\end{bmatrix} = \\theta\" alt=\"\\nabla\\left( - H(p) \\right) = \\begin{bmatrix} \\ln p_{1} + 1 &amp; \\ldots &amp; \\ln p_{n} + 1 \\end{bmatrix} = \\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fb5\/fb53dcbf243e93492acc764791a6e25e.svg\" width=\"344\" height=\"18\" data-width=\"42.952\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 42.952ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fb5\/fb53dcbf243e93492acc764791a6e25e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fb5\/fb53dcbf243e93492acc764791a6e25e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u043a\u0430\u043a\u043e\u0433\u043e-\u0442\u043e \u0443\u0441\u0442\u043e\u044f\u0432\u0448\u0435\u0433\u043e\u0441\u044f \u043d\u0430\u0437\u0432\u0430\u043d\u0438\u044f, \u043d\u043e \u0441 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e \u0434\u043e \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u044b \u2013 \u044d\u0442\u043e \u043b\u043e\u0433\u0438\u0442\u044b \/ \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0448\u0430\u043d\u0441\u044b: <img decoding=\"async\" class=\"formula inline\" source=\"\\theta_{i} = \\ln\\frac{p_{i}}{p_{base}} = \\ln p_{i} - \\ln p_{base}\" alt=\"\\theta_{i} = \\ln\\frac{p_{i}}{p_{base}} = \\ln p_{i} - \\ln p_{base}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c7\/3c758ac269f32a579e3eb4f2870f5326.svg\" width=\"232\" height=\"36\" data-width=\"28.974\" data-height=\"4.52\" data-vertical-align=\"-1.695\" style=\"width: 28.974ex; height: 4.52ex; vertical-align: -1.695ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c7\/3c758ac269f32a579e3eb4f2870f5326.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c7\/3c758ac269f32a579e3eb4f2870f5326.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u0440\u0438\u0447\u0435\u043c \u0442\u043e\u0447\u043d\u043e\u0435 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u043f\u0440\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"p_{base} = \\frac{1}{e}\" alt=\"p_{base} = \\frac{1}{e}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f64\/f64efdab97b7996592c58ae9c12f0179.svg\" width=\"76\" height=\"37\" data-width=\"9.498\" data-height=\"4.613\" data-vertical-align=\"-1.741\" style=\"width: 9.498ex; height: 4.613ex; vertical-align: -1.741ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f64\/f64efdab97b7996592c58ae9c12f0179.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f64\/f64efdab97b7996592c58ae9c12f0179.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u0438 \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u043c, \u0447\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\nabla^{2}\\left( - H(p) \\right) = \\begin{pmatrix} \\frac{1}{p_{1}} &amp; \\cdots &amp; 0 \\\\ \\vdots &amp; \\ddots &amp; \\vdots \\\\ 0 &amp; \\cdots &amp; \\frac{1}{p_{n}} \\end{pmatrix} \\succ 0\" alt=\"\\nabla^{2}\\left( - H(p) \\right) = \\begin{pmatrix} \\frac{1}{p_{1}} &amp; \\cdots &amp; 0 \\\\ \\vdots &amp; \\ddots &amp; \\vdots \\\\ 0 &amp; \\cdots &amp; \\frac{1}{p_{n}} \\end{pmatrix} \\succ 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/158\/158b2d48490c34b942b2dfef89fa00f0.svg\" width=\"283\" height=\"95\" data-width=\"35.337\" data-height=\"11.91\" data-vertical-align=\"-5.389\" style=\"width: 35.337ex; height: 11.91ex; vertical-align: -5.389ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/158\/158b2d48490c34b942b2dfef89fa00f0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/158\/158b2d48490c34b942b2dfef89fa00f0.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0413\u0435\u0441\u0441\u0438\u0430\u043d \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0441 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430\u043c\u0438, \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u043c\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044f\u043c. \u041f\u043e \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0438 \u0441 \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432\u043e\u0439 \u043f\u0430\u0440\u043e\u0439 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d-\u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0438\u0430\u043d \u0432 \u0444\u0438\u0437\u0438\u043a\u0435, \u0438 \u043a\u0430\u043a \u043c\u044b \u0432\u044b\u0432\u043e\u0434\u0438\u043b\u0438 \u043f\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0435 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043c\u043e\u0436\u043d\u043e \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u0442\u044c, \u0447\u0442\u043e \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043a\u0430\u043a-\u0442\u043e \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0441 \u043f\u0440\u044f\u043c\u044b\u043c\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044f\u043c\u0438. \u0411\u0443\u0434\u0435\u0442 \u043b\u0438 \u043e\u043d \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u043c \u2013 \u043d\u0435 \u0444\u0430\u043a\u0442, \u043d\u0430\u0434\u043e \u043f\u0440\u043e\u0432\u0435\u0440\u044f\u0442\u044c \u043f\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0435 \u043e\u0431 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/p>\n<p>\u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043e\u0442 \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\psi^{*}(\\theta) = \\sup_{p \\in \\Delta}\\left( \\left\\langle \\theta,p \\right\\rangle - \\sum_{i}^{}{p_{i}\\ln p_{i}} \\right)\" alt=\"\\psi^{*}(\\theta) = \\sup_{p \\in \\Delta}\\left( \\left\\langle \\theta,p \\right\\rangle - \\sum_{i}^{}{p_{i}\\ln p_{i}} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2d\/2db\/2dba9f99ee8e1e3b02534bbaf47347a0.svg\" width=\"268\" height=\"54\" data-width=\"33.438\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 33.438ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2d\/2db\/2dba9f99ee8e1e3b02534bbaf47347a0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2d\/2db\/2dba9f99ee8e1e3b02534bbaf47347a0.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e \u0437\u0430\u0434\u0430\u0447\u0430 \u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0439 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \u043d\u0430 \u0441\u0438\u043c\u043f\u043b\u0435\u043a\u0441\u0435. \u0420\u0435\u0448\u0430\u0442\u044c \u0443\u043c\u0435\u0435\u043c \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 \u0438\u0437 \u0440\u0430\u0437\u0434\u0435\u043b\u0430 \u043f\u0440\u043e \u043c\u0435\u0442\u043e\u0434\u044b \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"L = \\sum_{i}^{}{p_{i}\\theta_{i}} - \\sum_{i}^{}{p_{i}\\ln p_{i}} + \\lambda\\left( 1 - \\sum_{i}^{}p_{i} \\right),\\ \\ \\frac{\\partial L}{\\partial p_{i}} = \\theta_{i} - \\ln p_{i} - 1 - \\lambda = 0\\ \\\u00a0 \\Rightarrow \\ \\ p_{i} = e^{\\theta_{i} - 1 - \\lambda}\" alt=\"L = \\sum_{i}^{}{p_{i}\\theta_{i}} - \\sum_{i}^{}{p_{i}\\ln p_{i}} + \\lambda\\left( 1 - \\sum_{i}^{}p_{i} \\right),\\ \\ \\frac{\\partial L}{\\partial p_{i}} = \\theta_{i} - \\ln p_{i} - 1 - \\lambda = 0\\ \\\u00a0 \\Rightarrow \\ \\ p_{i} = e^{\\theta_{i} - 1 - \\lambda}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fec\/fec90bf92658a56a138bdea58d2d7d97.svg\" width=\"718\" height=\"54\" data-width=\"89.711\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 89.711ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fec\/fec90bf92658a56a138bdea58d2d7d97.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fe\/fec\/fec90bf92658a56a138bdea58d2d7d97.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u0442\u0434\u0435\u043b\u044c\u043d\u043e \u043e\u0442\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \u044d\u0442\u043e \u043f\u043e\u043b\u043d\u043e\u0441\u0442\u044c\u044e \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u0435\u0442 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430, \u0442.\u043a. \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u043f\u0440\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda\" alt=\"\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg\" width=\"12\" height=\"18\" data-width=\"1.319\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.319ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0437\u0430\u043d\u0443\u043b\u044f\u0435\u0442\u0441\u044f \u043d\u0430 \u0432\u0441\u0435\u0445 <img decoding=\"async\" class=\"formula inline\" source=\"p\" alt=\"p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg\" width=\"12\" height=\"12\" data-width=\"1.138\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.138ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u0437 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u043c\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430. \u0421\u0430\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda\" alt=\"\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg\" width=\"12\" height=\"18\" data-width=\"1.319\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.319ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e\u0434\u0431\u0438\u0440\u0430\u0435\u043c \u0438\u0437 \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\sum_{i}^{}p_{i} = 1\" alt=\"\\sum_{i}^{}p_{i} = 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b1\/6b1bf333aa5585db2888263842affbc6.svg\" width=\"77\" height=\"44\" data-width=\"9.67\" data-height=\"5.525\" data-vertical-align=\"-2.197\" style=\"width: 9.67ex; height: 5.525ex; vertical-align: -2.197ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b1\/6b1bf333aa5585db2888263842affbc6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b1\/6b1bf333aa5585db2888263842affbc6.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p_{i} = \\frac{e^{\\theta_{i}}}{\\sum_{i}^{}e^{\\theta_{i}}}\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ p = softmax\\ \\theta\" alt=\"p_{i} = \\frac{e^{\\theta_{i}}}{\\sum_{i}^{}e^{\\theta_{i}}}\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ p = softmax\\ \\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b0\/b0c\/b0c169cfd115953ef41d9eb7ad615fec.svg\" width=\"266\" height=\"47\" data-width=\"33.303\" data-height=\"5.833\" data-vertical-align=\"-2.351\" style=\"width: 33.303ex; height: 5.833ex; vertical-align: -2.351ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b0\/b0c\/b0c169cfd115953ef41d9eb7ad615fec.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b0\/b0c\/b0c169cfd115953ef41d9eb7ad615fec.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u043c \u0432 <img decoding=\"async\" class=\"formula inline\" source=\"\\psi^{*}\" alt=\"\\psi^{*}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg\" width=\"20\" height=\"18\" data-width=\"2.461\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.461ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\psi^{*}(\\theta) = \\sum_{i}^{}{\\theta_{i}p_{i}} - \\sum_{i}^{}{p_{i}\\ln p_{i}} = \\sum_{i}^{}{p_{i}\\theta_{i}} - \\sum_{i}^{}{p_{i}\\left( \\theta_{i} - \\ln{\\sum_{i}^{}e^{\\theta_{i}}} \\right) = \\sum_{i}^{}{p_{i}\\ln{\\sum_{i}^{}e^{\\theta_{i}}}}} = \\ln{\\sum_{i}^{}e^{\\theta_{i}}}\" alt=\"\\psi^{*}(\\theta) = \\sum_{i}^{}{\\theta_{i}p_{i}} - \\sum_{i}^{}{p_{i}\\ln p_{i}} = \\sum_{i}^{}{p_{i}\\theta_{i}} - \\sum_{i}^{}{p_{i}\\left( \\theta_{i} - \\ln{\\sum_{i}^{}e^{\\theta_{i}}} \\right) = \\sum_{i}^{}{p_{i}\\ln{\\sum_{i}^{}e^{\\theta_{i}}}}} = \\ln{\\sum_{i}^{}e^{\\theta_{i}}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd80b83d644ee2cf25ef911f2f31019b.svg\" width=\"742\" height=\"54\" data-width=\"92.785\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 92.785ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd80b83d644ee2cf25ef911f2f31019b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bd\/bd8\/bd80b83d644ee2cf25ef911f2f31019b.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"\\left( - H(p) \\right)^{*} = logsumexp(\\theta)\" alt=\"\\left( - H(p) \\right)^{*} = logsumexp(\\theta)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dc\/dce\/dceb7c8856236ea885741d88572b3439.svg\" width=\"207\" height=\"19\" data-width=\"25.932\" data-height=\"2.389\" data-vertical-align=\"-0.629\" style=\"width: 25.932ex; height: 2.389ex; vertical-align: -0.629ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dc\/dce\/dceb7c8856236ea885741d88572b3439.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dc\/dce\/dceb7c8856236ea885741d88572b3439.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u043e\u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c, \u0447\u0442\u043e\u0431\u044b \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u0442\u044c \u043d\u0430\u0448\u0438 \u0434\u043e\u0433\u0430\u0434\u043a\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\nabla logsumexp(\\theta) = \\begin{bmatrix} \\frac{e^{\\theta_{1}}}{\\sum_{i}^{}e^{\\theta_{i}}} &amp; \\ldots &amp; \\frac{e^{\\theta_{n}}}{\\sum_{i}^{}e^{\\theta_{i}}} \\end{bmatrix} = softmax\\ \\theta = p\" alt=\"\\nabla logsumexp(\\theta) = \\begin{bmatrix} \\frac{e^{\\theta_{1}}}{\\sum_{i}^{}e^{\\theta_{i}}} &amp; \\ldots &amp; \\frac{e^{\\theta_{n}}}{\\sum_{i}^{}e^{\\theta_{i}}} \\end{bmatrix} = softmax\\ \\theta = p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/59\/597\/59738ccec2ff410c0ddd7b109188b15a.svg\" width=\"448\" height=\"33\" data-width=\"55.945\" data-height=\"4.07\" data-vertical-align=\"-1.469\" style=\"width: 55.945ex; height: 4.07ex; vertical-align: -1.469ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/59\/597\/59738ccec2ff410c0ddd7b109188b15a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/59\/597\/59738ccec2ff410c0ddd7b109188b15a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442 logsumexp \u0443\u0436\u0435 \u0438\u043c\u0435\u0435\u0442 \u0443\u0441\u0442\u043e\u044f\u0432\u0448\u0435\u0435\u0441\u044f \u043d\u0430\u0437\u0432\u0430\u043d\u0438\u0435, \u0441\u043e\u0431\u0441\u0442\u0432\u0435\u043d\u043d\u043e, softmax. \u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0433\u0435\u0441\u0441\u0438\u0430\u043d:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\nabla^{2}logsumexp(\\theta) = \\backslash \u0433\u0440\u043e\u043c\u043e\u0437\u0434\u043a\u0430\u044f\\ \u0432\u044b\u043a\u043b\u0430\u0434\u043a\u0430\\backslash\\text{=}diag(p) - pp^{T} = \\begin{pmatrix} p_{1} - p_{1}^{2} &amp; \\cdots &amp; - p_{1}p_{n} \\\\ \\vdots &amp; \\ddots &amp; \\vdots \\\\ - p_{n}p_{1} &amp; \\cdots &amp; p_{n} - p_{n}^{2} \\end{pmatrix} \\succcurlyeq 0\" alt=\"\\nabla^{2}logsumexp(\\theta) = \\backslash \u0433\u0440\u043e\u043c\u043e\u0437\u0434\u043a\u0430\u044f\\ \u0432\u044b\u043a\u043b\u0430\u0434\u043a\u0430\\backslash\\text{=}diag(p) - pp^{T} = \\begin{pmatrix} p_{1} - p_{1}^{2} &amp; \\cdots &amp; - p_{1}p_{n} \\\\ \\vdots &amp; \\ddots &amp; \\vdots \\\\ - p_{n}p_{1} &amp; \\cdots &amp; p_{n} - p_{n}^{2} \\end{pmatrix} \\succcurlyeq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/39\/395\/395516258281736125e05796bb294952.svg\" width=\"758\" height=\"87\" data-width=\"94.803\" data-height=\"10.816\" data-vertical-align=\"-4.842\" style=\"width: 94.803ex; height: 10.816ex; vertical-align: -4.842ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/39\/395\/395516258281736125e05796bb294952.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/39\/395\/395516258281736125e05796bb294952.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0413\u0435\u0441\u0441\u0438\u0430\u043d \u0432\u044b\u0440\u043e\u0436\u0434\u0435\u043d, \u043e\u0431\u0440\u0430\u0442\u0438\u043c\u043e\u0441\u0442\u0438 \u043d\u0435\u0442 (\u0441\u0438\u043c\u043f\u043b\u0435\u043a\u0441 \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\mathbb{R}^{n}\" alt=\"\\mathbb{R}^{n}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f65\/f65c4f70e25cf0c782818fa85ab12bb7.svg\" width=\"22\" height=\"18\" data-width=\"2.781\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.781ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f65\/f65c4f70e25cf0c782818fa85ab12bb7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f6\/f65\/f65c4f70e25cf0c782818fa85ab12bb7.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u043c\u0435\u044e\u0442 \u0440\u0430\u0437\u043d\u0443\u044e \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c: \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043a\u0438 \u00ab\u0441\u044a\u0435\u0434\u0430\u0435\u0442\u00bb \u043e\u0434\u043d\u0443 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0441\u0432\u043e\u0431\u043e\u0434\u044b). \u041d\u043e \u043c\u044b \u043f\u043e\u0447\u0442\u0438 \u0443\u0433\u0430\u0434\u0430\u043b\u0438, \u0447\u0442\u043e \u0432\u043e\u0437\u043d\u0438\u043a\u043d\u0443\u0442 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 (\u043f\u0440\u0430\u0432\u0434\u0430, \u0441 \u043a\u043e\u043c\u0431\u0438\u043d\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u0439 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c\u044e).<\/p>\n<p>\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0432\u0441\u0435 \u043d\u0430\u0448\u0438 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f \u043c\u043e\u0436\u043d\u043e \u043e\u0431\u043e\u0431\u0449\u0438\u0442\u044c \u043d\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b\u0439 \u0441\u043b\u0443\u0447\u0430\u0439:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\psi(p) = - H\\lbrack p\\rbrack = \\int_{}^{}{p(x)\\ln{p(x)}dx},\\ \\ \\nabla\\left( - H\\lbrack p\\rbrack \\right) = \\ln{p(x)} + 1,\\ \\ \\nabla^{2}\\left( - H\\lbrack p\\rbrack \\right) = \\frac{\\delta(x - y)}{p(x)}\" alt=\"\\psi(p) = - H\\lbrack p\\rbrack = \\int_{}^{}{p(x)\\ln{p(x)}dx},\\ \\ \\nabla\\left( - H\\lbrack p\\rbrack \\right) = \\ln{p(x)} + 1,\\ \\ \\nabla^{2}\\left( - H\\lbrack p\\rbrack \\right) = \\frac{\\delta(x - y)}{p(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/da\/da6\/da6fd77059ec3fd109f91031d480600b.svg\" width=\"671\" height=\"44\" data-width=\"83.876\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 83.876ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/da\/da6\/da6fd77059ec3fd109f91031d480600b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/da\/da6\/da6fd77059ec3fd109f91031d480600b.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u043a\u0430\u043a \u0433\u0435\u0441\u0441\u0438\u0430\u043d &#8212; \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0439 \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440, \u0432\u043c\u0435\u0441\u0442\u043e \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u044c\u043d\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0441 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u043c\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044f\u043c\u0438.<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\psi^{*}(\\theta) = \\ln{\\int_{}^{}{e^{\\theta(x)}dx}},\\ \\ \\nabla\\psi^{*}(\\theta) = p(x),\\ \\ \\nabla^{2}\\psi^{*}(\\theta) = p(x)\\delta(x - y) - p(x)p(y)\" alt=\"\\psi^{*}(\\theta) = \\ln{\\int_{}^{}{e^{\\theta(x)}dx}},\\ \\ \\nabla\\psi^{*}(\\theta) = p(x),\\ \\ \\nabla^{2}\\psi^{*}(\\theta) = p(x)\\delta(x - y) - p(x)p(y)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/16\/169\/169b5a6612a3b1a7b83f2d00b6d9b9b2.svg\" width=\"584\" height=\"41\" data-width=\"72.955\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 72.955ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/16\/169\/169b5a6612a3b1a7b83f2d00b6d9b9b2.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/16\/169\/169b5a6612a3b1a7b83f2d00b6d9b9b2.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u043a\u0430\u043a \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 &#8212; \u0441\u0430\u043c\u0443 \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u044c, \u0430 \u043a\u0430\u043a \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u2013 \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440, \u043f\u043e\u0445\u043e\u0436\u0438\u0439 \u043f\u043e \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u0435 \u043d\u0430 \u043c\u0430\u0442\u0440\u0438\u0446\u0443 \u0432\u044b\u0448\u0435<\/p>\n<p>\u041f\u0440\u0438\u043d\u0446\u0438\u043f \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u0438 \u041e\u041c\u041f<\/p>\n<p>\u0412 \u0431\u043b\u043e\u043a\u0435 \u043f\u0440\u043e \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u043c\u044b \u043f\u0440\u043e\u0432\u0435\u043b\u0438 \u0442\u0430\u043a\u043e\u0435 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\psi^{*}(\\theta) = \\sup_{p \\in \\Delta}\\left( \\left\\langle \\theta,p \\right\\rangle - \\sum_{i}^{}{p_{i}\\ln p_{i}} \\right)\" alt=\"\\psi^{*}(\\theta) = \\sup_{p \\in \\Delta}\\left( \\left\\langle \\theta,p \\right\\rangle - \\sum_{i}^{}{p_{i}\\ln p_{i}} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2d\/2db\/2dba9f99ee8e1e3b02534bbaf47347a0.svg\" width=\"268\" height=\"54\" data-width=\"33.438\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 33.438ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2d\/2db\/2dba9f99ee8e1e3b02534bbaf47347a0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2d\/2db\/2dba9f99ee8e1e3b02534bbaf47347a0.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0418 \u043e\u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c, \u0447\u0442\u043e \u0441\u0432\u044f\u0437\u0430\u043d\u043d\u0430\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u0430\u0440\u0430 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0445 \u2013 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"p\" alt=\"p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg\" width=\"12\" height=\"12\" data-width=\"1.138\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.138ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0427\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442 \u0441\u0430\u043c \u0441\u0443\u043f\u0440\u0435\u043c\u0443\u043c: \u043c\u044b \u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u043b\u0438 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u043f\u0440\u0438 \u0444\u0438\u043a\u0441\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u043c \u043c\u0430\u0442\u043e\u0436\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\mathbb{E}_{p}\\theta = \\left\\langle \\theta,p \\right\\rangle = \\sum_{i}^{}{p_{i}\\theta_{i}}\" alt=\"\\mathbb{E}_{p}\\theta = \\left\\langle \\theta,p \\right\\rangle = \\sum_{i}^{}{p_{i}\\theta_{i}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/110\/1107cc4b249a86395d45c1a656b54ca0.svg\" width=\"175\" height=\"44\" data-width=\"21.885\" data-height=\"5.525\" data-vertical-align=\"-2.197\" style=\"width: 21.885ex; height: 5.525ex; vertical-align: -2.197ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/110\/1107cc4b249a86395d45c1a656b54ca0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/110\/1107cc4b249a86395d45c1a656b54ca0.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412 \u044d\u0442\u043e\u043c \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u0438, softmax \u2013 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0439 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u043a \u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u0434\u043e\u0439\u0442\u0438 \u0441 \u0434\u0440\u0443\u0433\u043e\u0439 \u0441\u0442\u043e\u0440\u043e\u043d\u044b, \u0438 \u043d\u0430\u0447\u043d\u0435\u043c \u0441 \u043f\u0430\u0440\u044b \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u044b\u0445 \u0437\u0430\u0434\u0430\u0447.<\/p>\n<p>\u041f\u0440\u0438\u043c\u0435\u0440 1: \u043c\u043e\u043d\u0435\u0442\u043a\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"X_{i}\\ \\sim\\ Bernoulli(\\mu)\" alt=\"X_{i}\\ \\sim\\ Bernoulli(\\mu)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f3b\/f3b18dc39f48bec03aab1be1d32c59cc.svg\" width=\"156\" height=\"18\" data-width=\"19.555\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 19.555ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f3b\/f3b18dc39f48bec03aab1be1d32c59cc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f3b\/f3b18dc39f48bec03aab1be1d32c59cc.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u0434\u0431\u0440\u043e\u0441\u0438\u043c \u0435\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"n\" alt=\"n\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b8\/7b8b965ad4bca0e41ab51de7b31363a1.svg\" width=\"12\" height=\"12\" data-width=\"1.357\" data-height=\"1.025\" data-vertical-align=\"-0.025\" style=\"width: 1.357ex; height: 1.025ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b8\/7b8b965ad4bca0e41ab51de7b31363a1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b8\/7b8b965ad4bca0e41ab51de7b31363a1.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0440\u0430\u0437, \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0434\u043e\u043b\u044e \u043f\u043e\u044f\u0432\u043b\u0435\u043d\u0438\u0439 \u0440\u0435\u0448\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"m = \\frac{1}{n}\\sum_{}^{}x_{i}\" alt=\"m = \\frac{1}{n}\\sum_{}^{}x_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/441\/441f0de149bac46048499208d24be8f6.svg\" width=\"107\" height=\"45\" data-width=\"13.411\" data-height=\"5.638\" data-vertical-align=\"-2.253\" style=\"width: 13.411ex; height: 5.638ex; vertical-align: -2.253ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/441\/441f0de149bac46048499208d24be8f6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/441\/441f0de149bac46048499208d24be8f6.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041a\u0430\u043a \u043e\u0446\u0435\u043d\u0438\u0442\u044c \u043f\u043e \u043d\u0430\u0448\u0438\u043c \u0434\u0430\u043d\u043d\u044b\u043c, \u043a\u0430\u043a\u043e\u0435 \u0440\u0435\u0430\u043b\u044c\u043d\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u043a\u043e\u0440\u0435\u0435 \u0432\u0441\u0435\u0433\u043e \u043c\u043e\u0433\u043b\u043e \u044d\u0442\u0438 \u0434\u0430\u043d\u043d\u044b\u0435 \u043f\u043e\u0440\u043e\u0434\u0438\u0442\u044c?<\/p>\n<p>\u0424\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e, \u043c\u044b \u0438\u0449\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\mu} = \\arg{\\max_{\\mu}{p\\left( x \\middle| \\mu \\right)}} = \\arg{\\max_{\\mu}{\\ln\\left( p\\left( x \\middle| \\mu \\right) \\right)}}\" alt=\"\\widehat{\\mu} = \\arg{\\max_{\\mu}{p\\left( x \\middle| \\mu \\right)}} = \\arg{\\max_{\\mu}{\\ln\\left( p\\left( x \\middle| \\mu \\right) \\right)}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e62\/e62bc70c67e995cd489807cc63f5ee39.svg\" width=\"326\" height=\"31\" data-width=\"40.776\" data-height=\"3.855\" data-vertical-align=\"-1.362\" style=\"width: 40.776ex; height: 3.855ex; vertical-align: -1.362ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e62\/e62bc70c67e995cd489807cc63f5ee39.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e62\/e62bc70c67e995cd489807cc63f5ee39.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442.\u043a. \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0441\u0442\u0440\u043e\u0433\u043e \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u0435\u0442 \u0438 \u043d\u0435 \u0441\u0434\u0432\u0438\u0433\u0430\u0435\u0442 \u0430\u0440\u0433\u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c. \u0412\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"p\\left( x \\middle| \\mu \\right)\" alt=\"p\\left( x \\middle| \\mu \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/29\/299\/299f987c43fca17fc2eb92b3b3ccf27e.svg\" width=\"53\" height=\"18\" data-width=\"6.563\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.563ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/29\/299\/299f987c43fca17fc2eb92b3b3ccf27e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/29\/299\/299f987c43fca17fc2eb92b3b3ccf27e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u0435\u043c, \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"l(\\mu) = \\ln\\left( p\\left( x \\middle| \\mu \\right) \\right)\" alt=\"l(\\mu) = \\ln\\left( p\\left( x \\middle| \\mu \\right) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e3\/e3d\/e3dc82a290694b0304783272d356ec85.svg\" width=\"139\" height=\"18\" data-width=\"17.402\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 17.402ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e3\/e3d\/e3dc82a290694b0304783272d356ec85.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e3\/e3d\/e3dc82a290694b0304783272d356ec85.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u043c \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f, \u0435\u0435 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u2013 \u0441\u043a\u043e\u0440\u043e\u043c, \u0430 \u0441\u0430\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\mu}\" alt=\"\\widehat{\\mu}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/54\/54f\/54ff0e1154ea2a35a612f93a5dbd7ea9.svg\" width=\"12\" height=\"20\" data-width=\"1.364\" data-height=\"2.466\" data-vertical-align=\"-0.667\" style=\"width: 1.364ex; height: 2.466ex; vertical-align: -0.667ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/54\/54f\/54ff0e1154ea2a35a612f93a5dbd7ea9.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/54\/54f\/54ff0e1154ea2a35a612f93a5dbd7ea9.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043e\u0446\u0435\u043d\u043a\u043e\u0439 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f (\u041e\u041c\u041f). \u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"l(\\mu) = \\ln{p\\left( x \\middle| \\mu \\right)} = \\ln{\\prod_{i = 1}^{n}{p\\left( x_{i} \\middle| \\mu \\right)}} = \\sum_{i = 1}^{n}{\\ln{p\\left( x_{i} \\middle| \\mu \\right)}} = \\sum_{i = 1}^{n}{\\ln\\left( \\mu^{x_{i}}(1 - \\mu)^{1 - x_{i}} \\right)} = \\ln\\mu \\cdot \\sum_{}^{}x_{i} + \\ln(1 - \\mu) \\cdot \\left( n - \\sum_{}^{}x_{i} \\right) = mn\\ln\\mu + (n - mn)\\ln(1 - \\mu)\" alt=\"l(\\mu) = \\ln{p\\left( x \\middle| \\mu \\right)} = \\ln{\\prod_{i = 1}^{n}{p\\left( x_{i} \\middle| \\mu \\right)}} = \\sum_{i = 1}^{n}{\\ln{p\\left( x_{i} \\middle| \\mu \\right)}} = \\sum_{i = 1}^{n}{\\ln\\left( \\mu^{x_{i}}(1 - \\mu)^{1 - x_{i}} \\right)} = \\ln\\mu \\cdot \\sum_{}^{}x_{i} + \\ln(1 - \\mu) \\cdot \\left( n - \\sum_{}^{}x_{i} \\right) = mn\\ln\\mu + (n - mn)\\ln(1 - \\mu)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c5\/3c5e6a4bb700474da4daeccc0b5046ab.svg\" width=\"1158\" height=\"54\" data-width=\"144.696\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 144.696ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c5\/3c5e6a4bb700474da4daeccc0b5046ab.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3c\/3c5\/3c5e6a4bb700474da4daeccc0b5046ab.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"0 = \\frac{d}{d\\mu}l(\\mu) = \\frac{m}{\\mu}n - \\frac{1 - m}{1 - \\mu}n\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\frac{\\widehat{\\mu}}{1 - \\widehat{\\mu}} = \\frac{m}{1 - m}\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ \\ \\widehat{\\mu} = m\" alt=\"0 = \\frac{d}{d\\mu}l(\\mu) = \\frac{m}{\\mu}n - \\frac{1 - m}{1 - \\mu}n\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\frac{\\widehat{\\mu}}{1 - \\widehat{\\mu}} = \\frac{m}{1 - m}\\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ \\ \\widehat{\\mu} = m\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/67\/67e\/67e18052d7db60f68660b64d67437164.svg\" width=\"530\" height=\"46\" data-width=\"66.188\" data-height=\"5.729\" data-vertical-align=\"-2.299\" style=\"width: 66.188ex; height: 5.729ex; vertical-align: -2.299ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/67\/67e\/67e18052d7db60f68660b64d67437164.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/67\/67e\/67e18052d7db60f68660b64d67437164.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u0430\u0438\u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u043d\u0430 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044f \u2013 \u043a\u043e\u0433\u0434\u0430 \u043c\u043e\u043d\u0435\u0442\u043a\u0430 \u0432\u044b\u043f\u0430\u0434\u0430\u0435\u0442 \u0440\u0435\u0448\u043a\u043e\u0439 \u0441 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c\u044e, \u0440\u0430\u0432\u043d\u043e\u0439 \u043d\u0430\u0448\u0435\u0439 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u043e \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u043d\u043d\u043e\u0439 \u0434\u043e\u043b\u0435 \u0440\u0435\u0448\u0435\u043a, \u0447\u0442\u043e \u043b\u043e\u0433\u0438\u0447\u043d\u043e, \u0435\u0441\u043b\u0438 \u043e \u043c\u043e\u043d\u0435\u0442\u043a\u0435 \u043d\u0435 \u0434\u043e\u043f\u0443\u0441\u043a\u0430\u0442\u044c \u043d\u0438\u0447\u0435\u0433\u043e \u0434\u043e\u043f\u043e\u043b\u043d\u0438\u0442\u0435\u043b\u044c\u043d\u043e.<\/p>\n<p>\u0417\u0430\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \u0432 \u0432\u044b\u0432\u043e\u0434\u0435 \u0443 \u043d\u0430\u0441 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u043b\u0430 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\widehat{\\mu}}{1 - \\widehat{\\mu}}\" alt=\"\\frac{\\widehat{\\mu}}{1 - \\widehat{\\mu}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e4\/e48\/e48a26a913ccb00e4383ec457c2d2981.svg\" width=\"50\" height=\"46\" data-width=\"6.257\" data-height=\"5.729\" data-vertical-align=\"-2.299\" style=\"width: 6.257ex; height: 5.729ex; vertical-align: -2.299ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e4\/e48\/e48a26a913ccb00e4383ec457c2d2981.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e4\/e48\/e48a26a913ccb00e4383ec457c2d2981.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043b\u044e\u0441 \u043c\u044b \u0432 \u043d\u0430\u0447\u0430\u043b\u0435 \u0440\u0430\u0437\u0434\u0435\u043b\u0430 \u0437\u0430\u043c\u0435\u0442\u0438\u043b\u0438, \u0447\u0442\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0443\u0447\u0430\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u0439 \u0432 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0439 \u043f\u0430\u0440\u0435, \u0431\u044b\u043b \u043b\u043e\u0433\u0438\u0442\u043e\u043c. \u0425\u043e\u0447\u0435\u0442\u0441\u044f \u043f\u043e\u0434\u043e\u0433\u043d\u0430\u0442\u044c \u0437\u0430\u0434\u0430\u0447\u0443 \u043f\u043e\u0434 \u0443\u0436\u0435 \u0437\u043d\u0430\u043a\u043e\u043c\u044b\u0439 \u043d\u0430\u043c \u0432\u0438\u0434: \u0432\u043e\u0437\u044c\u043c\u0435\u043c \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta = \\ln\\frac{\\mu}{1 - \\mu}\" alt=\"\\theta = \\ln\\frac{\\mu}{1 - \\mu}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6e\/6eb\/6eb581497e7157aa646bed2e170b0fd5.svg\" width=\"101\" height=\"37\" data-width=\"12.599\" data-height=\"4.57\" data-vertical-align=\"-1.719\" style=\"width: 12.599ex; height: 4.57ex; vertical-align: -1.719ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6e\/6eb\/6eb581497e7157aa646bed2e170b0fd5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6e\/6eb\/6eb581497e7157aa646bed2e170b0fd5.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta} = \\ln\\frac{m}{1 - m}\" alt=\"\\widehat{\\theta} = \\ln\\frac{m}{1 - m}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d0\/d00\/d007640a758c27a8880d8f353fc2e8af.svg\" width=\"106\" height=\"34\" data-width=\"13.221\" data-height=\"4.267\" data-vertical-align=\"-1.568\" style=\"width: 13.221ex; height: 4.267ex; vertical-align: -1.568ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d0\/d00\/d007640a758c27a8880d8f353fc2e8af.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d0\/d00\/d007640a758c27a8880d8f353fc2e8af.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041a\u0430\u043a \u0431\u044b \u043c\u044b \u0440\u0435\u0448\u0430\u043b\u0438 \u044d\u0442\u0443 \u0437\u0430\u0434\u0430\u0447\u0443 \u0440\u0430\u043d\u044c\u0448\u0435, \u043c\u0435\u0442\u043e\u0434\u043e\u043c \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0439 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438? \u041f\u043e\u0438\u0449\u0435\u043c \u0442\u0430\u043a\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"p(x)\" alt=\"p(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/41\/413\/4130c89f2d12c3ac81aba3adbff28685.svg\" width=\"34\" height=\"18\" data-width=\"4.192\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.192ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/41\/413\/4130c89f2d12c3ac81aba3adbff28685.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/41\/413\/4130c89f2d12c3ac81aba3adbff28685.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0434\u0430\u0435\u0442 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u0443\u044e \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u043f\u0440\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u0438, \u0447\u0442\u043e \u0435\u0433\u043e \u043c\u0430\u0442\u043e\u0436\u0438\u0434\u0430\u043d\u0438\u0435 \u0440\u0430\u0432\u043d\u043e \u043d\u0430\u0431\u043b\u044e\u0434\u0430\u0435\u043c\u043e\u043c\u0443 \u0441\u0440\u0435\u0434\u043d\u0435\u043c\u0443: <img decoding=\"async\" class=\"formula inline\" source=\"\\mathbb{E}_{p}X = \\sum p(x)x = p(1) = m\" alt=\"\\mathbb{E}_{p}X = \\sum p(x)x = p(1) = m\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/89\/897\/89792a57231dcbd2c8a9509c40ed2af6.svg\" width=\"229\" height=\"25\" data-width=\"28.627\" data-height=\"3.167\" data-vertical-align=\"-1.018\" style=\"width: 28.627ex; height: 3.167ex; vertical-align: -1.018ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/89\/897\/89792a57231dcbd2c8a9509c40ed2af6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/89\/897\/89792a57231dcbd2c8a9509c40ed2af6.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0417\u0430\u043f\u0438\u0448\u0435\u043c \u0437\u0430\u0434\u0430\u0447\u0443 \u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0439 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438, \u043e\u043d\u0430 \u0436\u0435 \u2013 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439 (\u0434\u0432\u0430 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044f \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 &#8212; <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda\" alt=\"\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg\" width=\"12\" height=\"18\" data-width=\"1.319\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.319ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"L = - \\sum_{}^{}{p\\ln p} + \\theta\\left( \\sum_{}^{}{px} - m \\right) + \\lambda\\left( 1 - \\sum_{}^{}p \\right),\\ \\ 0 = \\frac{\\partial L}{\\partial p} = - \\ln p - 1 - \\theta x - \\lambda\\ \\ \\\u00a0 \\Rightarrow \\ \\ p(x) \\propto e^{\\theta x}\" alt=\"L = - \\sum_{}^{}{p\\ln p} + \\theta\\left( \\sum_{}^{}{px} - m \\right) + \\lambda\\left( 1 - \\sum_{}^{}p \\right),\\ \\ 0 = \\frac{\\partial L}{\\partial p} = - \\ln p - 1 - \\theta x - \\lambda\\ \\ \\\u00a0 \\Rightarrow \\ \\ p(x) \\propto e^{\\theta x}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f0\/f03\/f03af1e2ff7baed6e36d1437cd5b3e0d.svg\" width=\"791\" height=\"54\" data-width=\"98.822\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 98.822ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f0\/f03\/f03af1e2ff7baed6e36d1437cd5b3e0d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f0\/f03\/f03af1e2ff7baed6e36d1437cd5b3e0d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"p(1) = m = \\frac{e^{\\widehat{\\theta} \\cdot 1}}{e^{\\widehat{\\theta} \\cdot 1} + e^{\\widehat{\\theta} \\cdot 0}} = \\frac{1}{1 + e^{- \\widehat{\\theta}}} = \\sigma(\\widehat{\\theta})\" alt=\"p(1) = m = \\frac{e^{\\widehat{\\theta} \\cdot 1}}{e^{\\widehat{\\theta} \\cdot 1} + e^{\\widehat{\\theta} \\cdot 0}} = \\frac{1}{1 + e^{- \\widehat{\\theta}}} = \\sigma(\\widehat{\\theta})\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/906\/906bf041d7c4e798b3ceef9d764e19ae.svg\" width=\"324\" height=\"52\" data-width=\"40.562\" data-height=\"6.533\" data-vertical-align=\"-2.701\" style=\"width: 40.562ex; height: 6.533ex; vertical-align: -2.701ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/906\/906bf041d7c4e798b3ceef9d764e19ae.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/906\/906bf041d7c4e798b3ceef9d764e19ae.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f, \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u043b\u043e\u0433\u0438\u0442\u0443, \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0441\u0438\u0433\u043c\u043e\u0438\u0434\u043e\u0439. \u0414\u043b\u044f \u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0430\u043b\u0438 \u043f\u043e\u0445\u043e\u0436\u0438\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 &#8212; softmax. \u0418\u0437 \u044d\u0442\u043e\u0433\u043e \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta} = \\ln\\frac{m}{1 - m}\" alt=\"\\widehat{\\theta} = \\ln\\frac{m}{1 - m}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d0\/d00\/d007640a758c27a8880d8f353fc2e8af.svg\" width=\"106\" height=\"34\" data-width=\"13.221\" data-height=\"4.267\" data-vertical-align=\"-1.568\" style=\"width: 13.221ex; height: 4.267ex; vertical-align: -1.568ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d0\/d00\/d007640a758c27a8880d8f353fc2e8af.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d0\/d00\/d007640a758c27a8880d8f353fc2e8af.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0440\u043e\u0432\u043d\u043e \u0442\u0430\u043a\u043e\u0439 \u0436\u0435, \u043a\u0430\u043a \u0434\u043b\u044f \u041e\u041c\u041f. \u041a \u0442\u043e\u043c\u0443 \u0436\u0435, \u0443\u043a\u0440\u0435\u043f\u0438\u043b\u0430\u0441\u044c \u043d\u0430\u0448\u0430 \u0438\u043d\u0442\u0443\u0438\u0446\u0438\u044f \u043e \u0437\u0430\u043f\u0438\u0441\u0438 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 \u0432 \u0432\u0438\u0434\u0435 \u043b\u043e\u0433\u0438\u0442\u0430.<\/p>\n<p>\u041f\u0440\u0438\u043c\u0435\u0440 2: \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"X_{i}\\ \\sim\\ N\\left( \\mu,\\sigma^{2} \\right),\\ \\ m_{1} = \\frac{1}{n}\\sum_{}^{}x_{i},\\ \\ m_{2}^{2} = \\frac{1}{n}\\sum_{}^{}x_{i}^{2}\" alt=\"X_{i}\\ \\sim\\ N\\left( \\mu,\\sigma^{2} \\right),\\ \\ m_{1} = \\frac{1}{n}\\sum_{}^{}x_{i},\\ \\ m_{2}^{2} = \\frac{1}{n}\\sum_{}^{}x_{i}^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/eb\/ebb\/ebbd3912befd0f6a6077ba2e7e6325f9.svg\" width=\"394\" height=\"45\" data-width=\"49.19\" data-height=\"5.638\" data-vertical-align=\"-2.253\" style=\"width: 49.19ex; height: 5.638ex; vertical-align: -2.253ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/eb\/ebb\/ebbd3912befd0f6a6077ba2e7e6325f9.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/eb\/ebb\/ebbd3912befd0f6a6077ba2e7e6325f9.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0422\u0435\u043f\u0435\u0440\u044c \u0443 \u043d\u0430\u0441 2 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 + 2 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u044b \u0441\u0447\u0438\u0442\u0430\u0435\u043c \u0432 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0435. \u041e\u0431\u043e\u0437\u043d\u0430\u0447\u0438\u043c \u0432\u0435\u043a\u0442\u043e\u0440 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a \u043a\u0430\u043a <img decoding=\"async\" class=\"formula inline\" source=\"T(x) = \\begin{pmatrix} x &amp; x^{2} \\end{pmatrix}^{T}\" alt=\"T(x) = \\begin{pmatrix} x &amp; x^{2} \\end{pmatrix}^{T}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f53\/f534fdee2c3a4918673b1398cc5a8d68.svg\" width=\"133\" height=\"23\" data-width=\"16.577\" data-height=\"2.918\" data-vertical-align=\"-0.893\" style=\"width: 16.577ex; height: 2.918ex; vertical-align: -0.893ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f53\/f534fdee2c3a4918673b1398cc5a8d68.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f53\/f534fdee2c3a4918673b1398cc5a8d68.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u041c\u041f:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"l\\left( \\mu,\\sigma^{2} \\right) = - \\frac{n}{2}\\ln\\left( 2\\pi\\sigma^{2} \\right) - \\frac{1}{2\\sigma^{2}}\\sum_{}^{}\\left( x_{i} - \\mu \\right)^{2},\\ \\ \\partial_{\\mu} = \\partial_{\\sigma^{2}} = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\begin{pmatrix} \\widehat{\\mu} \\\\ {\\widehat{\\sigma}}^{2} \\end{pmatrix} = \\begin{pmatrix} m_{1} \\\\ m_{2}^{2} - m_{1}^{2} \\end{pmatrix}\" alt=\"l\\left( \\mu,\\sigma^{2} \\right) = - \\frac{n}{2}\\ln\\left( 2\\pi\\sigma^{2} \\right) - \\frac{1}{2\\sigma^{2}}\\sum_{}^{}\\left( x_{i} - \\mu \\right)^{2},\\ \\ \\partial_{\\mu} = \\partial_{\\sigma^{2}} = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\begin{pmatrix} \\widehat{\\mu} \\\\ {\\widehat{\\sigma}}^{2} \\end{pmatrix} = \\begin{pmatrix} m_{1} \\\\ m_{2}^{2} - m_{1}^{2} \\end{pmatrix}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ee\/ee7\/ee7501c79f9092279be3564fcf2b2168.svg\" width=\"684\" height=\"49\" data-width=\"85.533\" data-height=\"6.086\" data-vertical-align=\"-2.477\" style=\"width: 85.533ex; height: 6.086ex; vertical-align: -2.477ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ee\/ee7\/ee7501c79f9092279be3564fcf2b2168.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ee\/ee7\/ee7501c79f9092279be3564fcf2b2168.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u0438\u043d\u0446\u0438\u043f \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 (\u043f\u0440\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 <img decoding=\"async\" class=\"formula inline\" source=\"\\mathbb{E}_{p}X = m_{1},\\ \\ \\mathbb{E}_{p}X^{2} = m_{2}^{2}\" alt=\"\\mathbb{E}_{p}X = m_{1},\\ \\ \\mathbb{E}_{p}X^{2} = m_{2}^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d5\/d55\/d55814552d9184c9ee4322a991adf799.svg\" width=\"193\" height=\"21\" data-width=\"24.081\" data-height=\"2.65\" data-vertical-align=\"-0.759\" style=\"width: 24.081ex; height: 2.65ex; vertical-align: -0.759ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d5\/d55\/d55814552d9184c9ee4322a991adf799.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d5\/d55\/d55814552d9184c9ee4322a991adf799.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0441\u043e \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044f\u043c\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta_{1},\\theta_{2},\\lambda\" alt=\"\\theta_{1},\\theta_{2},\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d16\/d163e6bb4838c3a21d8c1e907eea97e0.svg\" width=\"59\" height=\"18\" data-width=\"7.429\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.429ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d16\/d163e6bb4838c3a21d8c1e907eea97e0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d16\/d163e6bb4838c3a21d8c1e907eea97e0.svg 781w\" loading=\"lazy\" decode=\"async\"\/>):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"L = - \\int_{}^{}{p\\ln p} + \\theta_{1}\\left( m_{1} - \\int_{}^{}{pxdx} \\right) + \\theta_{2}\\left( m_{2}^{2} - \\int_{}^{}{px^{2}dx} \\right) + \\lambda\\left( 1 - \\int_{}^{}{pdx} \\right)\" alt=\"L = - \\int_{}^{}{p\\ln p} + \\theta_{1}\\left( m_{1} - \\int_{}^{}{pxdx} \\right) + \\theta_{2}\\left( m_{2}^{2} - \\int_{}^{}{px^{2}dx} \\right) + \\lambda\\left( 1 - \\int_{}^{}{pdx} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d45\/d45823eebcd6c4c2df435c4a7477ec6d.svg\" width=\"621\" height=\"43\" data-width=\"77.657\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 77.657ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d45\/d45823eebcd6c4c2df435c4a7477ec6d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d45\/d45823eebcd6c4c2df435c4a7477ec6d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"0 = \\frac{\\delta L}{\\delta p} = - \\ln p - 1 - \\theta_{1}x - \\theta_{2}x^{2} - \\lambda\\ \\ \\\u00a0 \\Rightarrow \\ \\ p(x) \\propto e^{\\theta_{1}x + \\theta_{2}x^{2}}\" alt=\"0 = \\frac{\\delta L}{\\delta p} = - \\ln p - 1 - \\theta_{1}x - \\theta_{2}x^{2} - \\lambda\\ \\ \\\u00a0 \\Rightarrow \\ \\ p(x) \\propto e^{\\theta_{1}x + \\theta_{2}x^{2}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a8\/3a8b29b637d51bacb7d9e56b0a42c1db.svg\" width=\"471\" height=\"41\" data-width=\"58.901\" data-height=\"5.143\" data-vertical-align=\"-2.006\" style=\"width: 58.901ex; height: 5.143ex; vertical-align: -2.006ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a8\/3a8b29b637d51bacb7d9e56b0a42c1db.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a8\/3a8b29b637d51bacb7d9e56b0a42c1db.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0417\u043d\u0430\u0447\u0438\u0442, <img decoding=\"async\" class=\"formula inline\" source=\"p(x)\" alt=\"p(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/41\/413\/4130c89f2d12c3ac81aba3adbff28685.svg\" width=\"34\" height=\"18\" data-width=\"4.192\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.192ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/41\/413\/4130c89f2d12c3ac81aba3adbff28685.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/41\/413\/4130c89f2d12c3ac81aba3adbff28685.svg 781w\" loading=\"lazy\" decode=\"async\"\/> &#8212; \u0433\u0430\u0443\u0441\u0441\u0438\u0430\u043d\u0430, \u0430 \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\begin{pmatrix} {\\widehat{\\theta}}_{1} &amp; {\\widehat{\\theta}}_{2} \\end{pmatrix} = \\begin{pmatrix} \\frac{\\widehat{\\mu}}{{\\widehat{\\sigma}}^{2}} &amp; - \\frac{1}{2{\\widehat{\\sigma}}^{2}} \\end{pmatrix}\" alt=\"\\begin{pmatrix} {\\widehat{\\theta}}_{1} &amp; {\\widehat{\\theta}}_{2} \\end{pmatrix} = \\begin{pmatrix} \\frac{\\widehat{\\mu}}{{\\widehat{\\sigma}}^{2}} &amp; - \\frac{1}{2{\\widehat{\\sigma}}^{2}} \\end{pmatrix}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b3\/8b372b3393fa4a8fb4f9c017288abc77.svg\" width=\"194\" height=\"33\" data-width=\"24.188\" data-height=\"4.07\" data-vertical-align=\"-1.469\" style=\"width: 24.188ex; height: 4.07ex; vertical-align: -1.469ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b3\/8b372b3393fa4a8fb4f9c017288abc77.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b3\/8b372b3393fa4a8fb4f9c017288abc77.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u0430 \u044d\u0442\u0438\u0445 \u043f\u0440\u0438\u043c\u0435\u0440\u0430\u0445 \u043e\u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c, \u0447\u0442\u043e \u043c\u0435\u0442\u043e\u0434 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0439 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u0438 \u041e\u041c\u041f \u2013 \u043f\u0440\u044f\u043c\u0430\u044f \u0438 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432\u043e \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u0438. \u0412\u0441\u0435\u0433\u0434\u0430 \u043b\u0438 \u044d\u0442\u043e \u0442\u0430\u043a?<\/p>\n<p>\u042d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u044b\u0439 \u043a\u043b\u0430\u0441\u0441, \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u0424\u0438\u0448\u0435\u0440\u0430 \u0438 \u043a\u043e\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u044f<\/p>\n<p>\u0427\u0442\u043e \u043e\u0431\u0449\u0435\u0433\u043e \u043c\u043e\u0436\u043d\u043e \u0437\u0430\u043c\u0435\u0442\u0438\u0442\u044c \u0432 \u043e\u0431\u043e\u0438\u0445 \u043f\u0440\u0438\u043c\u0435\u0440\u0430\u0445:<\/p>\n<ol>\n<li>\n<p>\u043c\u044b \u043d\u0435 \u0440\u0430\u0431\u043e\u0442\u0430\u043b\u0438 \u0441 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u043c\u0438 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u0430\u043c\u0438 \u0432\u044b\u0431\u043e\u0440\u043a\u0438. \u041f\u0440\u043e\u0432\u043e\u0434\u044f \u044d\u0442\u0438 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u044b \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435, \u043c\u044b \u043c\u043e\u0433\u043b\u0438 \u0432\u043e\u043e\u0431\u0449\u0435 \u043d\u0435 \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u0442\u044c \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u0435 \u0438\u0441\u0445\u043e\u0434\u044b. \u041d\u0430\u043c \u0431\u044b\u043b\u043e \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u043b\u0438\u0448\u044c \u0445\u0440\u0430\u043d\u0438\u0442\u044c 1-2 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u0438 \u043e\u0431\u043d\u043e\u0432\u043b\u044f\u0442\u044c \u0438\u0445 \u0438\u0442\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u043e \u2013 \u0438 \u043a\u0430\u043a \u0431\u0443\u0434\u0442\u043e \u0432 \u044d\u0442\u0438\u0445 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0430\u0445 \u0431\u044b\u043b\u043e \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u0438, \u0447\u0442\u043e\u0431\u044b \u043e\u043f\u0438\u0441\u0430\u0442\u044c \u0432\u0441\u0451 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435.<\/p>\n<\/li>\n<li>\n<p>\u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438 \u0432\u044b\u0440\u0430\u0436\u0430\u043b\u0438\u0441\u044c \u0447\u0435\u0440\u0435\u0437 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u0443, \u0435\u0441\u043b\u0438 \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u0442\u044c \u0438\u0445 \u0447\u0435\u0440\u0435\u0437 \u043f\u0440\u0430\u0432\u0438\u043b\u044c\u043d\u044b\u0435 (\u0441 \u0442\u043e\u0447\u043a\u0438 \u0437\u0440\u0435\u043d\u0438\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430) \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b. \u0418\u043c\u0435\u043d\u043d\u043e \u044d\u0442\u0438 \u00ab\u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435\u00bb \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u044b \u0441 \u043d\u0430\u0431\u043b\u044e\u0434\u0430\u0435\u043c\u044b\u043c\u0438 \u0441\u0440\u0435\u0434\u043d\u0438\u043c\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f\u043c\u0438 \u0442\u0435\u0445 \u00ab\u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u044b\u0445\u00bb \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a<\/p>\n<\/li>\n<\/ol>\n<p>\u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0443\u0434\u043e\u0431\u043d\u044b\u0439 \u043a\u043b\u0430\u0441\u0441 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0439, \u043e\u0431\u043e\u0431\u0449\u0430\u044e\u0449\u0438\u0439 \u043d\u0430\u0448\u0438 \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u044f \u2013 \u043e\u043d \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u044b\u043c. \u0412\u0441\u0435 \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438 \u0432 \u043d\u0435\u043c \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0442\u0441\u044f \u043a\u0430\u043a:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p\\left( x \\middle| \\theta \\right) = h(x)\\exp{\\left( \\theta T(x) - f(\\theta) \\right),\\ \\ \u0433\u0434\u0435\\ f(\\theta) = \\ln{\\int_{}^{}{h(x)\\exp{\\left( \\theta T(x) \\right)dx}}}}\" alt=\"p\\left( x \\middle| \\theta \\right) = h(x)\\exp{\\left( \\theta T(x) - f(\\theta) \\right),\\ \\ \u0433\u0434\u0435\\ f(\\theta) = \\ln{\\int_{}^{}{h(x)\\exp{\\left( \\theta T(x) \\right)dx}}}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/71\/71a\/71ae3c2d498b604c6df506d517c3ad6f.svg\" width=\"557\" height=\"41\" data-width=\"69.632\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 69.632ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/71\/71a\/71ae3c2d498b604c6df506d517c3ad6f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/71\/71a\/71ae3c2d498b604c6df506d517c3ad6f.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u0430\u0437\u0432\u0430\u043d\u0438\u044f \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442: <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440, <img decoding=\"async\" class=\"formula inline\" source=\"T(x)\" alt=\"T(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/aef\/aef077a1c7b98f6091af987a5eb78742.svg\" width=\"37\" height=\"18\" data-width=\"4.647\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.647ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/aef\/aef077a1c7b98f6091af987a5eb78742.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/aef\/aef077a1c7b98f6091af987a5eb78742.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u0430\u044f \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0430, <img decoding=\"async\" class=\"formula inline\" source=\"h(x)\" alt=\"h(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/ca8\/ca8e608169b20a94570ac837e8ba0833.svg\" width=\"35\" height=\"18\" data-width=\"4.357\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.357ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/ca8\/ca8e608169b20a94570ac837e8ba0833.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/ca8\/ca8e608169b20a94570ac837e8ba0833.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0431\u0430\u0437\u043e\u0432\u0430\u044f \u043c\u0435\u0440\u0430. \u041f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u0438 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442 \u043b\u0438\u043d\u0435\u0439\u043d\u044b \u043f\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0430 \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043a\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"f(\\theta)\" alt=\"f(\\theta)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/881\/8819683a72852fa92e5b0312071d4eb0.svg\" width=\"33\" height=\"18\" data-width=\"4.066\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.066ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/881\/8819683a72852fa92e5b0312071d4eb0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/88\/881\/8819683a72852fa92e5b0312071d4eb0.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u043c\u0435\u0435\u0442 \u0437\u043d\u0430\u043a\u043e\u043c\u044b\u0439 \u043d\u0430\u043c \u0432\u0438\u0434 logsumexp \u0438 \u0432\u044b\u043f\u0443\u043a\u043b\u0430 (\u0432\u043d\u0435 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043a\u043b\u0430\u0441\u0441\u0430, \u043a\u0441\u0442\u0430\u0442\u0438, \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u044c f \u043d\u0435 \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u0443\u0435\u0442\u0441\u044f):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\nabla f = \\mathbb{E}_{\\theta}T(X) = \\mu,\\ \\ \\nabla^{2}f = Var\\ T(X) \\geq 0\" alt=\"\\nabla f = \\mathbb{E}_{\\theta}T(X) = \\mu,\\ \\ \\nabla^{2}f = Var\\ T(X) \\geq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/40\/407\/4072fd94a38181f22427c9036dd28deb.svg\" width=\"332\" height=\"21\" data-width=\"41.478\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 41.478ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/40\/407\/4072fd94a38181f22427c9036dd28deb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/40\/407\/4072fd94a38181f22427c9036dd28deb.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u0438 \u0440\u0430\u0441\u0447\u0435\u0442\u0435 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043a\u0430\u043a \u0440\u0430\u0437 \u043f\u043e\u043b\u0435\u0437\u0435\u043d \u0442\u0435\u043c, \u0447\u0442\u043e \u00ab\u0441\u044a\u0435\u0434\u0430\u0435\u0442\u00bb \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u0443. \u0414\u043b\u044f 1 \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\frac{1}{n}l(\\theta) = const + \\theta\\widehat{\\mu} - f(\\theta),\\ \\ \u0433\u0434\u0435\\ \\widehat{\\mu} = \\frac{1}{n}\\sum_{}^{}{T\\left( x_{i} \\right)}\" alt=\"\\frac{1}{n}l(\\theta) = const + \\theta\\widehat{\\mu} - f(\\theta),\\ \\ \u0433\u0434\u0435\\ \\widehat{\\mu} = \\frac{1}{n}\\sum_{}^{}{T\\left( x_{i} \\right)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b0\/8b04242cdf56116aadb1ba87e7a47ff3.svg\" width=\"396\" height=\"45\" data-width=\"49.495\" data-height=\"5.638\" data-vertical-align=\"-2.253\" style=\"width: 49.495ex; height: 5.638ex; vertical-align: -2.253ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b0\/8b04242cdf56116aadb1ba87e7a47ff3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b0\/8b04242cdf56116aadb1ba87e7a47ff3.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0421\u0440\u0430\u0437\u0443 \u0432\u0438\u0434\u043d\u043e, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\mathbb{E}\\nabla l(\\theta)\\mathbb{= E}T - \\mu = 0.\" alt=\"\\mathbb{E}\\nabla l(\\theta)\\mathbb{= E}T - \\mu = 0.\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b89\/b898aaca896d1a12fe81023ced9a0bbf.svg\" width=\"172\" height=\"18\" data-width=\"21.549\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 21.549ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b89\/b898aaca896d1a12fe81023ced9a0bbf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b8\/b89\/b898aaca896d1a12fe81023ced9a0bbf.svg 781w\" loading=\"lazy\" decode=\"async\"\/>\u0410 max \u043f\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f^{*}\\left( \\widehat{\\mu} \\right) = \\max_{\\theta}\\left( \\theta\\widehat{\\mu} - f(\\theta) \\right),\\ \\ \\theta = \\nabla f^{*}(\\mu) = (\\nabla f)^{- 1}(\\mu),\\ \\ \\nabla^{2}f^{*} = \\left( Var\\ T(X)\\\u00a0 \\right)^{- 1}\" alt=\"f^{*}\\left( \\widehat{\\mu} \\right) = \\max_{\\theta}\\left( \\theta\\widehat{\\mu} - f(\\theta) \\right),\\ \\ \\theta = \\nabla f^{*}(\\mu) = (\\nabla f)^{- 1}(\\mu),\\ \\ \\nabla^{2}f^{*} = \\left( Var\\ T(X)\\\u00a0 \\right)^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e67\/e67956bc23737497063e1294f3e29279.svg\" width=\"613\" height=\"31\" data-width=\"76.679\" data-height=\"3.918\" data-vertical-align=\"-1.393\" style=\"width: 76.679ex; height: 3.918ex; vertical-align: -1.393ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e67\/e67956bc23737497063e1294f3e29279.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e67\/e67956bc23737497063e1294f3e29279.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412 \u043c\u0435\u0442\u043e\u0434\u0435 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u0430 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u043c\u044b \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u043c \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u043f\u0440\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\mathbb{E}_{p}T = \\widehat{\\mu}\" alt=\"\\mathbb{E}_{p}T = \\widehat{\\mu}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/44e\/44e9de21f84e969f3e60f08d9f02110f.svg\" width=\"68\" height=\"20\" data-width=\"8.476\" data-height=\"2.55\" data-vertical-align=\"-0.709\" style=\"width: 8.476ex; height: 2.55ex; vertical-align: -0.709ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/44e\/44e9de21f84e969f3e60f08d9f02110f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/44e\/44e9de21f84e969f3e60f08d9f02110f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e\u043f\u0442\u0438\u043c\u0443\u043c \u0434\u0430\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"p(x) \\propto he^{\\theta T}\" alt=\"p(x) \\propto he^{\\theta T}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0f\/0f2\/0f2930401a728d11d53c3979ebe6a154.svg\" width=\"93\" height=\"21\" data-width=\"11.631\" data-height=\"2.628\" data-vertical-align=\"-0.748\" style=\"width: 11.631ex; height: 2.628ex; vertical-align: -0.748ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0f\/0f2\/0f2930401a728d11d53c3979ebe6a154.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0f\/0f2\/0f2930401a728d11d53c3979ebe6a154.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0432 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u043c \u043a\u043b\u0430\u0441\u0441\u0435, \u0430 \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0438 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u044b:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\min_{p}{KL\\left( p||h \\right)}\u043f\u0440\u0438\\ \\mathbb{E}_{p}T = \\widehat{\\mu}\" alt=\"\\min_{p}{KL\\left( p||h \\right)}\u043f\u0440\u0438\\ \\mathbb{E}_{p}T = \\widehat{\\mu}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/535\/53574d81465a60e3243dcff251e36f05.svg\" width=\"213\" height=\"30\" data-width=\"26.651\" data-height=\"3.795\" data-vertical-align=\"-1.332\" style=\"width: 26.651ex; height: 3.795ex; vertical-align: -1.332ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/535\/53574d81465a60e3243dcff251e36f05.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/535\/53574d81465a60e3243dcff251e36f05.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412 \u043d\u0430\u0447\u0430\u043b\u0435 \u0440\u0430\u0437\u0434\u0435\u043b\u0430 \u043c\u044b \u0440\u0430\u0431\u043e\u0442\u0430\u043b\u0438 \u0441 \u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u0430\u043b\u044c\u043d\u044b\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"p\" alt=\"p\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg\" width=\"12\" height=\"12\" data-width=\"1.138\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.138ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/83\/838\/83878c91171338902e0fe0fb97a8c47a.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0414\u0432\u0430 \u0432\u0430\u0436\u043d\u044b\u0445 \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u044f \u043e\u0442\u0442\u0443\u0434\u0430:<\/p>\n<ol>\n<li>\n<p>\u0433\u0435\u0441\u0441\u0438\u0430\u043d \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u043e\u043a\u0430\u0437\u0430\u043b\u0441\u044f \u0440\u0430\u0432\u0435\u043d \u0435\u0433\u043e \u043c\u0430\u0442\u0440\u0438\u0446\u0435 \u0424\u0438\u0448\u0435\u0440\u0430 (\u043f\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e, <img decoding=\"async\" class=\"formula inline\" source=\"I(\\theta) = Var\\ \\nabla l(\\theta)\\mathbb{= - E}\\nabla^{2}l(\\theta)\" alt=\"I(\\theta) = Var\\ \\nabla l(\\theta)\\mathbb{= - E}\\nabla^{2}l(\\theta)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cf\/cf6\/cf6ebe81f4f04590e96393918c99ddba.svg\" width=\"235\" height=\"21\" data-width=\"29.432\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 29.432ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cf\/cf6\/cf6ebe81f4f04590e96393918c99ddba.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cf\/cf6\/cf6ebe81f4f04590e96393918c99ddba.svg 781w\" loading=\"lazy\" decode=\"async\"\/>)<\/p>\n<\/li>\n<li>\n<p>\u0433\u0435\u0441\u0441\u0438\u0430\u043d logsumexp \u043e\u043a\u0430\u0437\u0430\u043b\u0441\u044f \u0440\u0430\u0432\u0435\u043d \u0435\u0433\u043e \u043c\u0430\u0442\u0440\u0438\u0446\u0435 \u043a\u043e\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u0438 (\u043f\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e, <img decoding=\"async\" class=\"formula inline\" source=\"Cov\\ X = \\mathbb{E}XX^{T}\\mathbb{- E}X\\mathbb{E}X^{T}\" alt=\"Cov\\ X = \\mathbb{E}XX^{T}\\mathbb{- E}X\\mathbb{E}X^{T}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8c8\/8c8fba6239a70d6d00b55553e7af204a.svg\" width=\"212\" height=\"21\" data-width=\"26.544\" data-height=\"2.583\" data-vertical-align=\"-0.726\" style=\"width: 26.544ex; height: 2.583ex; vertical-align: -0.726ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8c8\/8c8fba6239a70d6d00b55553e7af204a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8c8\/8c8fba6239a70d6d00b55553e7af204a.svg 781w\" loading=\"lazy\" decode=\"async\"\/>)<\/p>\n<\/li>\n<\/ol>\n<p>\u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u0442\u0430\u043a\u0436\u0435 \u0442\u0440\u0438 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b\u0445 \u043d\u0430\u043c \u0441\u043c\u044b\u0441\u043b\u0430 \u0433\u0435\u0441\u0441\u0438\u0430\u043d\u0430:<\/p>\n<ol>\n<li>\n<p>\u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439: \u0432\u0442\u043e\u0440\u0430\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430\u044f, \u043b\u043e\u043a\u0430\u043b\u044c\u043d\u0430\u044f \u043c\u0435\u0440\u0430 \u0432\u044b\u043f\u0443\u043a\u043b\u043e\u0441\u0442\u0438\/\u043a\u0440\u0438\u0432\u0438\u0437\u043d\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/p>\n<\/li>\n<li>\n<p>\u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043a\u0438\u0439: \u0430\u043d\u0430\u043b\u043e\u0433\u0438 \u043c\u0430\u0441\u0441\u044b\/\u0438\u043d\u0435\u0440\u0446\u0438\u0438 \u0438 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u043c \u043a \u043d\u0438\u043c<\/p>\n<\/li>\n<li>\n<p>\u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0439: \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u043a\u043e\u0434\u0438\u0440\u0443\u0435\u0442 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044e (\u043c\u0435\u0442\u0440\u0438\u043a\u0443) \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430, \u043d\u0430 \u043a\u043e\u0442\u043e\u0440\u043e\u043c \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f<\/p>\n<\/li>\n<\/ol>\n<p>\u0412\u044b\u0448\u0435 \u043c\u044b \u043f\u043e\u043a\u0430\u0437\u0430\u043b\u0438, \u043c\u0430\u0442\u0440\u0438\u0446\u0430 \u0424\u0438\u0448\u0435\u0440\u0430 (\u0432 \u0441\u0440\u0435\u0434\u043d\u0438\u0445 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u0445) \u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u0430 \u043a\u043e\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u0438 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e\u0439 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 (\u0432 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u0445) \u0441\u0432\u044f\u0437\u0430\u043d\u044b \u043a\u0430\u043a \u0433\u0435\u0441\u0441\u0438\u0430\u043d\u044b \u0432 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0438 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430. \u0418\u0437 \u043d\u0435\u0433\u043e \u0436\u0435 \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u041a\u0440\u0430\u043c\u0435\u0440\u0430-\u0420\u0430\u043e. \u0414\u043b\u044f \u043d\u0435\u0441\u043c\u0435\u0449\u0435\u043d\u043d\u043e\u0439 \u043e\u0446\u0435\u043d\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta}\" alt=\"\\widehat{\\theta}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg\" width=\"12\" height=\"24\" data-width=\"1.061\" data-height=\"2.993\" data-vertical-align=\"-0.931\" style=\"width: 1.061ex; height: 2.993ex; vertical-align: -0.931ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (<img decoding=\"async\" class=\"formula inline\" source=\"\\mathbb{E}\\widehat{\\theta} = \\theta\" alt=\"\\mathbb{E}\\widehat{\\theta} = \\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ee\/ee2\/ee2aace357d5b788344e147eff208ae4.svg\" width=\"53\" height=\"24\" data-width=\"6.648\" data-height=\"2.993\" data-vertical-align=\"-0.931\" style=\"width: 6.648ex; height: 2.993ex; vertical-align: -0.931ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ee\/ee2\/ee2aace357d5b788344e147eff208ae4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ee\/ee2\/ee2aace357d5b788344e147eff208ae4.svg 781w\" loading=\"lazy\" decode=\"async\"\/>) \u043f\u043e \u041a\u0411\u0428:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"1 = Cov\\ \\left( \\widehat{\\theta},\\nabla l(\\theta) \\right)^{2} \\leq Var\\ \\widehat{\\theta} \\cdot Var\\ \\nabla l(\\theta) = Var\\ \\widehat{\\theta} \\cdot I(\\theta)\\ \\\u00a0 \\Rightarrow \\ \\ Var\\ \\widehat{\\theta} \\geq {I(\\theta)}^{- 1}\" alt=\"1 = Cov\\ \\left( \\widehat{\\theta},\\nabla l(\\theta) \\right)^{2} \\leq Var\\ \\widehat{\\theta} \\cdot Var\\ \\nabla l(\\theta) = Var\\ \\widehat{\\theta} \\cdot I(\\theta)\\ \\\u00a0 \\Rightarrow \\ \\ Var\\ \\widehat{\\theta} \\geq {I(\\theta)}^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e0b\/e0b4b3e016fce06e5d55ec905487f010.svg\" width=\"597\" height=\"36\" data-width=\"74.675\" data-height=\"4.518\" data-vertical-align=\"-1.693\" style=\"width: 74.675ex; height: 4.518ex; vertical-align: -1.693ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e0b\/e0b4b3e016fce06e5d55ec905487f010.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e0b\/e0b4b3e016fce06e5d55ec905487f010.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u0430 \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e \u043d\u0438\u043a\u0430\u043a\u0430\u044f \u043d\u0435\u0441\u043c\u0435\u0449\u0435\u043d\u043d\u0430\u044f \u043e\u0446\u0435\u043d\u043a\u0430 \u043d\u0435 \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u0442\u043e\u0447\u043d\u0435\u0435 \u0433\u0440\u0430\u043d\u0438\u0446\u044b <img decoding=\"async\" class=\"formula inline\" source=\"{I(\\theta)}^{- 1}\" alt=\"{I(\\theta)}^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77d\/77d93fd603c57410be98dc91808ccb2c.svg\" width=\"50\" height=\"22\" data-width=\"6.194\" data-height=\"2.71\" data-vertical-align=\"-0.79\" style=\"width: 6.194ex; height: 2.71ex; vertical-align: -0.79ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77d\/77d93fd603c57410be98dc91808ccb2c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/77\/77d\/77d93fd603c57410be98dc91808ccb2c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0424\u0438\u0437\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u044f \u043a\u043e\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u0439 \u2013 \u043c\u0430\u0441\u0441\u044b, \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u0438 \u0424\u0438\u0448\u0435\u0440\u0430 \u2013 \u043f\u043e\u0434\u0432\u0438\u0436\u043d\u043e\u0441\u0442\u0438 (\u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0435 \u043c\u0430\u0441\u0441\u044b). \u041a\u0440\u0430\u043c\u0435\u0440-\u0420\u0430\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e \u0447\u0435\u043c \u0436\u0451\u0441\u0442\u0447\u0435 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0430 \u0441\u0432\u044f\u0437\u0430\u043d\u0430 \u0441 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u043e\u043c (\u0431\u043e\u043b\u044c\u0448\u0435 \u043c\u0430\u0441\u0441\u0430), \u0442\u0435\u043c \u043c\u0435\u043d\u044c\u0448\u0435 \u043f\u043e\u0434\u0432\u0438\u0436\u043d\u043e\u0441\u0442\u0438 \u0438 \u0442\u0435\u043c \u043c\u0435\u043d\u044c\u0448\u0435 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u00ab\u0434\u0440\u043e\u0436\u0438\u0442\u00bb, \u0430 \u0437\u043d\u0430\u0447\u0438\u0442 &#8212; \u0442\u0435\u043c \u0442\u043e\u0447\u043d\u0435\u0435 \u043c\u043e\u0436\u043d\u043e \u043e\u0446\u0435\u043d\u0438\u0442\u044c \u0435\u0435 \u0440\u0435\u0430\u043b\u044c\u043d\u044b\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043f\u043e \u0448\u0443\u043c\u043d\u044b\u043c \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u044f\u043c.<\/p>\n<p>\u041e\u0441\u0442\u0430\u043b\u043e\u0441\u044c \u043f\u043e\u0433\u043e\u0432\u043e\u0440\u0438\u0442\u044c \u043f\u0440\u043e \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432. \u0412 \u0440\u0430\u0437\u0434\u0435\u043b\u0435 \u043f\u0440\u043e \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044e \u043c\u044b \u043c\u043e\u0433\u043b\u0438 \u043f\u043e-\u0440\u0430\u0437\u043d\u043e\u043c\u0443 \u0432\u044b\u0431\u0438\u0440\u0430\u0442\u044c \u043c\u0435\u0442\u0440\u0438\u043a\u0443 M, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0432 \u043c\u0435\u0442\u043e\u0434\u0435 \u041d\u044c\u044e\u0442\u043e\u043d\u0430 \u043c\u044b \u0431\u0440\u0430\u043b\u0438 \u0433\u0435\u0441\u0441\u0438\u0430\u043d \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u0412 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0435 \u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u043c\u0435\u0442\u0440\u0438\u043a\u0430 \u2013 \u044d\u0442\u043e \u043a\u0430\u043a \u0440\u0430\u0437 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u0424\u0438\u0448\u0435\u0440\u0430, \u0442.\u0435. \u0442\u043e\u0442 \u0436\u0435 \u0433\u0435\u0441\u0441\u0438\u0430\u043d (\u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430 \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043a\u0438). \u0427\u0435\u0441\u0442\u043d\u044b\u0439 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u0441 \u0443\u0447\u0435\u0442\u043e\u043c \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u043c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\widetilde{\\nabla}L = {I(\\theta)}^{- 1}\\nabla L\" alt=\"\\widetilde{\\nabla}L = {I(\\theta)}^{- 1}\\nabla L\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/ae1\/ae1f2526a7369a9a4b99b824b5ae6596.svg\" width=\"128\" height=\"23\" data-width=\"16.062\" data-height=\"2.857\" data-vertical-align=\"-0.863\" style=\"width: 16.062ex; height: 2.857ex; vertical-align: -0.863ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/ae1\/ae1f2526a7369a9a4b99b824b5ae6596.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/ae1\/ae1f2526a7369a9a4b99b824b5ae6596.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0432 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442, \u043c\u044b \u0437\u0430\u043f\u0443\u0441\u043a\u0430\u0435\u043c \u0432 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u043c\u0435\u0442\u043e\u0434 \u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0441\u043f\u0443\u0441\u043a\u0430 \u0441 \u0437\u0435\u0440\u043a\u0430\u043b\u043e\u043c \u2013 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u043c \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043a\u0438. \u042d\u0442\u043e\u0442 \u043c\u0435\u0442\u043e\u0434 \u0430\u043a\u0442\u0438\u0432\u043d\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442\u0441\u044f \u0432 \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u043c ML, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u043f\u043e\u043f\u0443\u043b\u044f\u0440\u043d\u044b\u0439 \u043c\u0435\u0442\u043e\u0434 Adam \u0430\u043f\u043f\u0440\u043e\u043a\u0441\u0438\u043c\u0438\u0440\u0443\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"I(\\theta)\" alt=\"I(\\theta)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/113\/113e8a4a1fd4a91fd6b3664411cbfc5f.svg\" width=\"32\" height=\"18\" data-width=\"3.962\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.962ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/113\/113e8a4a1fd4a91fd6b3664411cbfc5f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/113\/113e8a4a1fd4a91fd6b3664411cbfc5f.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u044c\u043d\u043e\u0439 \u043c\u0430\u0442\u0440\u0438\u0446\u0435\u0439 (\u043f\u043b\u044e\u0441 \u043f\u0430\u0440\u0430 \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u043d\u044b\u0445 \u0434\u043e\u0440\u0430\u0431\u043e\u0442\u043e\u043a). \u041f\u043e\u0434\u0440\u043e\u0431\u043d\u0435\u0435 \u043f\u0440\u043e \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u0432 ML \u043f\u043e\u0433\u043e\u0432\u043e\u0440\u0438\u043c \u043f\u043e\u0437\u0436\u0435.<\/p>\n<p>\u041f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u0433\u0438\u043f\u043e\u0442\u0435\u0437<\/p>\n<p>\u0412\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043a \u0437\u0430\u0434\u0430\u0447\u0435 \u0441 \u043c\u043e\u043d\u0435\u0442\u043a\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"X_{i}\\ \\sim\\ Bernoulli(\\mu)\" alt=\"X_{i}\\ \\sim\\ Bernoulli(\\mu)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f3b\/f3b18dc39f48bec03aab1be1d32c59cc.svg\" width=\"156\" height=\"18\" data-width=\"19.555\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 19.555ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f3b\/f3b18dc39f48bec03aab1be1d32c59cc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f3b\/f3b18dc39f48bec03aab1be1d32c59cc.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u0443\u0441\u0442\u044c \u043d\u0430\u043c \u0443\u0434\u0430\u043b\u043e\u0441\u044c \u0441\u0443\u0437\u0438\u0442\u044c \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0435 \u043c\u043e\u0434\u0435\u043b\u0438 \u0434\u043e \u0434\u0432\u0443\u0445 \u0430\u043b\u044c\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432: <img decoding=\"async\" class=\"formula inline\" source=\"H_{0}:X\\ \\sim\\ P_{0}\" alt=\"H_{0}:X\\ \\sim\\ P_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a7\/a76\/a76a0fe8f78295b3111db38adbd8b007.svg\" width=\"106\" height=\"18\" data-width=\"13.27\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.27ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a7\/a76\/a76a0fe8f78295b3111db38adbd8b007.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a7\/a76\/a76a0fe8f78295b3111db38adbd8b007.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043c\u043e\u043d\u0435\u0442\u043a\u0430 \u0447\u0435\u0441\u0442\u043d\u0430\u044f (<img decoding=\"async\" class=\"formula inline\" source=\"\\mu = \\frac{1}{2}\" alt=\"\\mu = \\frac{1}{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/865\/86580a2792403e4a686a32692f4ba19a.svg\" width=\"52\" height=\"37\" data-width=\"6.508\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 6.508ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/865\/86580a2792403e4a686a32692f4ba19a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/86\/865\/86580a2792403e4a686a32692f4ba19a.svg 781w\" loading=\"lazy\" decode=\"async\"\/>) vs <img decoding=\"async\" class=\"formula inline\" source=\"H_{1}:X\\ \\sim\\ P_{1}\" alt=\"H_{1}:X\\ \\sim\\ P_{1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/74\/748\/748dd49a2b23a704566ceb9e3e050d2d.svg\" width=\"106\" height=\"18\" data-width=\"13.27\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.27ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/74\/748\/748dd49a2b23a704566ceb9e3e050d2d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/74\/748\/748dd49a2b23a704566ceb9e3e050d2d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043c\u043e\u043d\u0435\u0442\u043a\u0430 \u0444\u0430\u043b\u044c\u0448\u0438\u0432\u0430\u044f (<img decoding=\"async\" class=\"formula inline\" source=\"\\mu = q\" alt=\"\\mu = q\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ac\/aca\/aca262434c0bac627e3640546d130d99.svg\" width=\"43\" height=\"18\" data-width=\"5.422\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.422ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ac\/aca\/aca262434c0bac627e3640546d130d99.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ac\/aca\/aca262434c0bac627e3640546d130d99.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u041d\u0443\u0436\u043d\u043e \u043f\u043e \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u044f\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c, \u0444\u0430\u043b\u044c\u0448\u0438\u0432\u0430\u044f \u043b\u0438 \u043c\u043e\u043d\u0435\u0442\u043a\u0430, \u0438 \u0447\u0435\u043c \u043c\u044b \u0440\u0438\u0441\u043a\u0443\u0435\u043c, \u043f\u0440\u0438\u043d\u0438\u043c\u0430\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u0435. \u0412\u0432\u0435\u0434\u0435\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"\\phi(x)\" alt=\"\\phi(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/16\/163\/163cde00287e629f33dae509a8414505.svg\" width=\"35\" height=\"18\" data-width=\"4.403\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 4.403ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/16\/163\/163cde00287e629f33dae509a8414505.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/16\/163\/163cde00287e629f33dae509a8414505.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c \u043e\u0442\u0432\u0435\u0440\u0433\u043d\u0443\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"H_{0}\" alt=\"H_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg\" width=\"23\" height=\"15\" data-width=\"2.868\" data-height=\"1.92\" data-vertical-align=\"-0.375\" style=\"width: 2.868ex; height: 1.92ex; vertical-align: -0.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"x\" alt=\"x\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg\" width=\"12\" height=\"12\" data-width=\"1.294\" data-height=\"1.025\" data-vertical-align=\"-0.025\" style=\"width: 1.294ex; height: 1.025ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<div>\n<div class=\"table\">\n<table>\n<tbody>\n<tr>\n<th>\n<p align=\"left\">\u041e\u0448\u0438\u0431\u043a\u0438<\/p>\n<\/th>\n<th>\n<p align=\"left\"><img decoding=\"async\" class=\"formula inline\" source=\"H_{0}\" alt=\"H_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg\" width=\"23\" height=\"15\" data-width=\"2.868\" data-height=\"1.92\" data-vertical-align=\"-0.375\" style=\"width: 2.868ex; height: 1.92ex; vertical-align: -0.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u0435\u0440\u043d\u0430<\/p>\n<\/th>\n<th>\n<p align=\"left\"><img decoding=\"async\" class=\"formula inline\" source=\"H_{1}\" alt=\"H_{1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7f\/7fc\/7fc2922e5316a1012606a2e9c181c4af.svg\" width=\"23\" height=\"15\" data-width=\"2.868\" data-height=\"1.885\" data-vertical-align=\"-0.339\" style=\"width: 2.868ex; height: 1.885ex; vertical-align: -0.339ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7f\/7fc\/7fc2922e5316a1012606a2e9c181c4af.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7f\/7fc\/7fc2922e5316a1012606a2e9c181c4af.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u0435\u0440\u043d\u0430<\/p>\n<\/th>\n<\/tr>\n<tr>\n<td>\n<p align=\"left\">\u041e\u0442\u0432\u0435\u0440\u0433\u0430\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"H_{0}\" alt=\"H_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg\" width=\"23\" height=\"15\" data-width=\"2.868\" data-height=\"1.92\" data-vertical-align=\"-0.375\" style=\"width: 2.868ex; height: 1.92ex; vertical-align: -0.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (<img decoding=\"async\" class=\"formula inline\" source=\"\\phi = 1\" alt=\"\\phi = 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/735\/7350d6ada4f3f8177ac6b58f33b3f04c.svg\" width=\"44\" height=\"18\" data-width=\"5.497\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.497ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/735\/7350d6ada4f3f8177ac6b58f33b3f04c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/735\/7350d6ada4f3f8177ac6b58f33b3f04c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>)<\/p>\n<\/td>\n<td>\n<p align=\"left\">I \u0440\u043e\u0434\u0430 (\u043b\u043e\u0436\u043d\u0430\u044f \u0442\u0440\u0435\u0432\u043e\u0433\u0430), <img decoding=\"async\" class=\"formula inline\" source=\"\\alpha = \\mathbb{E}_{0}\\phi\" alt=\"\\alpha = \\mathbb{E}_{0}\\phi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/703\/7035f3415d8cc704f3f6608e80f3f7a3.svg\" width=\"66\" height=\"18\" data-width=\"8.31\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.31ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/703\/7035f3415d8cc704f3f6608e80f3f7a3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/703\/7035f3415d8cc704f3f6608e80f3f7a3.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/td>\n<td>\n<p align=\"left\">\u0412\u0441\u0435 \u043e\u043a<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p align=\"left\">\u041d\u0435 \u043e\u0442\u0432\u0435\u0440\u0433\u0430\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"H_{0}\" alt=\"H_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg\" width=\"23\" height=\"15\" data-width=\"2.868\" data-height=\"1.92\" data-vertical-align=\"-0.375\" style=\"width: 2.868ex; height: 1.92ex; vertical-align: -0.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (<img decoding=\"async\" class=\"formula inline\" source=\"\\phi = 0\" alt=\"\\phi = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c96\/c96b1a3f13cad74e8bcd0a14c690dd42.svg\" width=\"44\" height=\"18\" data-width=\"5.497\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.497ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c96\/c96b1a3f13cad74e8bcd0a14c690dd42.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c96\/c96b1a3f13cad74e8bcd0a14c690dd42.svg 781w\" loading=\"lazy\" decode=\"async\"\/>)<\/p>\n<\/td>\n<td>\n<p align=\"left\">\u0412\u0441\u0435 \u043e\u043a<\/p>\n<\/td>\n<td>\n<p align=\"left\">II \u0440\u043e\u0434\u0430 (\u043f\u0440\u043e\u043f\u0443\u0441\u043a), <img decoding=\"async\" class=\"formula inline\" source=\"\\beta = \\mathbb{E}_{1}(1 - \\phi)\" alt=\"\\beta = \\mathbb{E}_{1}(1 - \\phi)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fc\/fc9\/fc9415d144fdc82d432c3ce77fd02c61.svg\" width=\"110\" height=\"18\" data-width=\"13.8\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.8ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fc\/fc9\/fc9415d144fdc82d432c3ce77fd02c61.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fc\/fc9\/fc9415d144fdc82d432c3ce77fd02c61.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>\u041d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u0432\u044b\u0431\u0438\u0440\u0430\u044e\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\alpha_{0}\" alt=\"\\alpha_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdb\/fdbc5fcfced7b0f661f8a414f6969ae5.svg\" width=\"19\" height=\"18\" data-width=\"2.436\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.436ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdb\/fdbc5fcfced7b0f661f8a414f6969ae5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdb\/fdbc5fcfced7b0f661f8a414f6969ae5.svg 781w\" loading=\"lazy\" decode=\"async\"\/> &#8212; \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u0439 \u0443\u0440\u043e\u0432\u0435\u043d\u044c \u043e\u0448\u0438\u0431\u043a\u0438 I \u0440\u043e\u0434\u0430 (\u0443\u0440\u043e\u0432\u0435\u043d\u044c \u0437\u043d\u0430\u0447\u0438\u043c\u043e\u0441\u0442\u0438), \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u0443\u044e\u0442, \u0447\u0442\u043e \u043b\u043e\u0436\u043d\u044b\u0445 \u0442\u0440\u0435\u0432\u043e\u0433 \u043d\u0435 \u0431\u043e\u043b\u044c\u0448\u0435 \u044d\u0442\u043e\u0433\u043e \u043f\u043e\u0440\u043e\u0433\u0430 (<img decoding=\"async\" class=\"formula inline\" source=\"\\alpha \\leq a_{0})\" alt=\"\\alpha \\leq a_{0})\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d24\/d245c65c4f3a00eb710e2dbc2e0dc0ec.svg\" width=\"60\" height=\"18\" data-width=\"7.53\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.53ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d24\/d245c65c4f3a00eb710e2dbc2e0dc0ec.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d2\/d24\/d245c65c4f3a00eb710e2dbc2e0dc0ec.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0438 \u0432 \u044d\u0442\u0438\u0445 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f\u0445 \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u044e\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\beta\" alt=\"\\beta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b0\/b06\/b0603860fcffe94e5b8eec59ed813421.svg\" width=\"12\" height=\"18\" data-width=\"1.281\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.281ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b0\/b06\/b0603860fcffe94e5b8eec59ed813421.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b0\/b06\/b0603860fcffe94e5b8eec59ed813421.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043e\u0448\u0438\u0431\u043a\u0443 II \u0440\u043e\u0434\u0430 (\u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u044e\u0442 \u043c\u043e\u0449\u043d\u043e\u0441\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"1 - \\beta\" alt=\"1 - \\beta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbc\/cbc32e6db60f2d35c9048c3dc9757292.svg\" width=\"41\" height=\"18\" data-width=\"5.177\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.177ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbc\/cbc32e6db60f2d35c9048c3dc9757292.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbc\/cbc32e6db60f2d35c9048c3dc9757292.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u042d\u0442\u043e \u0443\u0441\u043b\u043e\u0432\u043d\u0430\u044f \u043e\u043f\u0442\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044f:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"L(\\phi,\\lambda) = \\mathbb{E}_{1}\\phi - \\lambda\\left( \\mathbb{E}_{0}\\phi - \\alpha_{0} \\right) = \\lambda\\alpha_{0} + \\sum_{x}^{}{\\phi(x)\\left( P_{1}(x) - \\lambda P_{0}(x) \\right)}\" alt=\"L(\\phi,\\lambda) = \\mathbb{E}_{1}\\phi - \\lambda\\left( \\mathbb{E}_{0}\\phi - \\alpha_{0} \\right) = \\lambda\\alpha_{0} + \\sum_{x}^{}{\\phi(x)\\left( P_{1}(x) - \\lambda P_{0}(x) \\right)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/55\/559\/559b77d40dd05a704328839db359ac6e.svg\" width=\"511\" height=\"44\" data-width=\"63.9\" data-height=\"5.447\" data-vertical-align=\"-2.158\" style=\"width: 63.9ex; height: 5.447ex; vertical-align: -2.158ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/55\/559\/559b77d40dd05a704328839db359ac6e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/55\/559\/559b77d40dd05a704328839db359ac6e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u0430\u043a\u0441\u0438\u043c\u0443\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\max_{\\phi}L\" alt=\"\\max_{\\phi}L\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/37\/378\/3784446fe98456ee322be50b2ab17b0b.svg\" width=\"49\" height=\"30\" data-width=\"6.128\" data-height=\"3.764\" data-vertical-align=\"-1.316\" style=\"width: 6.128ex; height: 3.764ex; vertical-align: -1.316ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/37\/378\/3784446fe98456ee322be50b2ab17b0b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/37\/378\/3784446fe98456ee322be50b2ab17b0b.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u043e\u0437\u044c\u043c\u0435\u043c \u043f\u043e\u0442\u043e\u0447\u0435\u0447\u043d\u043e: \u043f\u043e\u0441\u0442\u0430\u0432\u0438\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\phi(x) = 1\" alt=\"\\phi(x) = 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f9\/f98\/f9867951e9e61a196f1b5649f8e21da5.svg\" width=\"68\" height=\"18\" data-width=\"8.551\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.551ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f9\/f98\/f9867951e9e61a196f1b5649f8e21da5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f9\/f98\/f9867951e9e61a196f1b5649f8e21da5.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0442\u0430\u043c, \u0433\u0434\u0435 \u0441\u043a\u043e\u0431\u043a\u0430 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u0430, \u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{P_{1}(x)}{P_{0}(x)} &gt; \\lambda\" alt=\"\\frac{P_{1}(x)}{P_{0}(x)} &gt; \\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/be7\/be79a0381ac9ae4fbbbbd32152f1be73.svg\" width=\"87\" height=\"44\" data-width=\"10.826\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 10.826ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/be7\/be79a0381ac9ae4fbbbbd32152f1be73.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/be\/be7\/be79a0381ac9ae4fbbbbd32152f1be73.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0417\u0430\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"P_{0}\" alt=\"P_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/13\/138\/13899aa46204b02713a854702142955d.svg\" width=\"20\" height=\"15\" data-width=\"2.44\" data-height=\"1.92\" data-vertical-align=\"-0.375\" style=\"width: 2.44ex; height: 1.92ex; vertical-align: -0.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/13\/138\/13899aa46204b02713a854702142955d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/13\/138\/13899aa46204b02713a854702142955d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"P_{1}\" alt=\"P_{1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/47\/47d\/47ddd0a8d1607438330cf19c0c1ac45e.svg\" width=\"20\" height=\"15\" data-width=\"2.44\" data-height=\"1.885\" data-vertical-align=\"-0.339\" style=\"width: 2.44ex; height: 1.885ex; vertical-align: -0.339ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/47\/47d\/47ddd0a8d1607438330cf19c0c1ac45e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/47\/47d\/47ddd0a8d1607438330cf19c0c1ac45e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f (\u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c \u0432\u044b\u0431\u043e\u0440\u043a\u0438 \u043f\u0440\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0445 \u0433\u0438\u043f\u043e\u0442\u0435\u0437 \u043d\u0430 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b). \u041c\u044b \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u043b\u0438 \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u0439 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u044f \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f \u043a\u0430\u043a \u0442\u043e\u0442, \u0447\u0442\u043e \u0434\u0430\u0435\u0442 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u0443\u044e \u043c\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u043f\u0440\u0438 \u0437\u0430\u0434\u0430\u043d\u043d\u043e\u043c \u0443\u0440\u043e\u0432\u043d\u0435 \u0437\u043d\u0430\u0447\u0438\u043c\u043e\u0441\u0442\u0438 \u2013 \u044d\u0442\u043e \u043b\u0435\u043c\u043c\u0430 \u041d\u0435\u0439\u043c\u0430\u043d\u0430-\u041f\u0438\u0440\u0441\u043e\u043d\u0430. \u041f\u0435\u0440\u0435\u0439\u0434\u0435\u043c \u043a \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0443 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"s(x) = \\ln\\frac{P_{1}(x)}{P_{0}(x)},\\ \\ S_{n} = \\frac{1}{n}\\sum_{}^{}{s(x_{i})},\\ \\ \u043e\u0442\u0432\u0435\u0440\u0433\u0430\u0435\u043c\\ H_{0}\\ \u043f\u0440\u0438\\ S_{n} &gt; \\gamma\" alt=\"s(x) = \\ln\\frac{P_{1}(x)}{P_{0}(x)},\\ \\ S_{n} = \\frac{1}{n}\\sum_{}^{}{s(x_{i})},\\ \\ \u043e\u0442\u0432\u0435\u0440\u0433\u0430\u0435\u043c\\ H_{0}\\ \u043f\u0440\u0438\\ S_{n} &gt; \\gamma\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/76e\/76e5b7251df034388556e61a99c19e1a.svg\" width=\"517\" height=\"47\" data-width=\"64.576\" data-height=\"5.905\" data-vertical-align=\"-2.387\" style=\"width: 64.576ex; height: 5.905ex; vertical-align: -2.387ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/76e\/76e5b7251df034388556e61a99c19e1a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/76e\/76e5b7251df034388556e61a99c19e1a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"S_{n}\" alt=\"S_{n}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/44d\/44d853a7808a331d95220fcb38095649.svg\" width=\"20\" height=\"16\" data-width=\"2.535\" data-height=\"1.952\" data-vertical-align=\"-0.357\" style=\"width: 2.535ex; height: 1.952ex; vertical-align: -0.357ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/44d\/44d853a7808a331d95220fcb38095649.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/44\/44d\/44d853a7808a331d95220fcb38095649.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u0430\u044f \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0430, \u043d\u0430\u043c \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u043a\u043e\u043f\u0438\u0442\u044c \u0442\u043e\u043b\u044c\u043a\u043e \u0435\u0435 \u0432\u043c\u0435\u0441\u0442\u043e \u0441\u044b\u0440\u044b\u0445 \u0434\u0430\u043d\u043d\u044b\u0445. \u041f\u043e \u0417\u0411\u0427:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\u041f\u043e\u0434\\ H_{0}:\\ \\mathbb{E}_{0}s = \\int_{}^{}{P_{0}(x)\\ln\\frac{P_{1}(x)}{P_{0}(x)}dx} = - KL(P_{0}|\\left| P_{1} \\right) \\leq 0\" alt=\"\u041f\u043e\u0434\\ H_{0}:\\ \\mathbb{E}_{0}s = \\int_{}^{}{P_{0}(x)\\ln\\frac{P_{1}(x)}{P_{0}(x)}dx} = - KL(P_{0}|\\left| P_{1} \\right) \\leq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1b\/1b5\/1b533d7b38185ceff8c79a547ebcc175.svg\" width=\"458\" height=\"44\" data-width=\"57.239\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 57.239ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1b\/1b5\/1b533d7b38185ceff8c79a547ebcc175.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1b\/1b5\/1b533d7b38185ceff8c79a547ebcc175.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"\u041f\u043e\u0434\\ H_{1}:\\ \\mathbb{E}_{1}s = \\int_{}^{}{P_{1}(x)\\ln\\frac{P_{1}(x)}{P_{0}(x)}dx} = KL(P_{1}|\\left| P_{0} \\right) \\geq 0\" alt=\"\u041f\u043e\u0434\\ H_{1}:\\ \\mathbb{E}_{1}s = \\int_{}^{}{P_{1}(x)\\ln\\frac{P_{1}(x)}{P_{0}(x)}dx} = KL(P_{1}|\\left| P_{0} \\right) \\geq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dd\/ddf\/ddf5e773207594c70f81bea7edb80a8d.svg\" width=\"444\" height=\"44\" data-width=\"55.479\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 55.479ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dd\/ddf\/ddf5e773207594c70f81bea7edb80a8d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dd\/ddf\/ddf5e773207594c70f81bea7edb80a8d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u0446\u0435\u043d\u0438\u043c \u043e\u0448\u0438\u0431\u043a\u0443 I \u0440\u043e\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\alpha_{n} = P_{0}\\left( S_{n} \\geq \\gamma \\right)\" alt=\"\\alpha_{n} = P_{0}\\left( S_{n} \\geq \\gamma \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/87\/87b\/87b44d8ff5b2da3351392f35ce105f14.svg\" width=\"136\" height=\"18\" data-width=\"16.97\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 16.97ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/87\/87b\/87b44d8ff5b2da3351392f35ce105f14.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/87\/87b\/87b44d8ff5b2da3351392f35ce105f14.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0414\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda \\geq 0\" alt=\"\\lambda \\geq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/43\/432\/432b4cfb979f14a140b7205d6bc5dc56.svg\" width=\"44\" height=\"18\" data-width=\"5.467\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.467ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/43\/432\/432b4cfb979f14a140b7205d6bc5dc56.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/43\/432\/432b4cfb979f14a140b7205d6bc5dc56.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u043d\u0434\u0438\u043a\u0430\u0442\u043e\u0440 \u043c\u0430\u0436\u043e\u0440\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u043e\u0439 (\u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u0427\u0435\u0440\u043d\u043e\u0432\u0430):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\alpha_{n} \\leq e^{- \\lambda n\\gamma}\\mathbb{E}_{0}\\left( e^{\\lambda\\sum_{}^{}{s\\left( x_{i} \\right)}} \\right) = e^{- \\lambda n\\gamma}\\left( \\mathbb{E}_{0}e^{\\lambda s} \\right)^{n} = e^{- n\\left( \\lambda\\gamma - \\psi(\\lambda) \\right)},\\ \\ \\psi(\\lambda) = \\ln{\\mathbb{E}_{0}\\left( e^{\\lambda s} \\right)} = \\ln{\\sum_{x}^{}{P_{0}(x)^{1 - \\lambda}P_{1}(x)^{\\lambda}}}\" alt=\"\\alpha_{n} \\leq e^{- \\lambda n\\gamma}\\mathbb{E}_{0}\\left( e^{\\lambda\\sum_{}^{}{s\\left( x_{i} \\right)}} \\right) = e^{- \\lambda n\\gamma}\\left( \\mathbb{E}_{0}e^{\\lambda s} \\right)^{n} = e^{- n\\left( \\lambda\\gamma - \\psi(\\lambda) \\right)},\\ \\ \\psi(\\lambda) = \\ln{\\mathbb{E}_{0}\\left( e^{\\lambda s} \\right)} = \\ln{\\sum_{x}^{}{P_{0}(x)^{1 - \\lambda}P_{1}(x)^{\\lambda}}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4de\/4de28e292e7290cc1808016c2e3f056b.svg\" width=\"795\" height=\"44\" data-width=\"99.32\" data-height=\"5.447\" data-vertical-align=\"-2.158\" style=\"width: 99.32ex; height: 5.447ex; vertical-align: -2.158ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4de\/4de28e292e7290cc1808016c2e3f056b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4d\/4de\/4de28e292e7290cc1808016c2e3f056b.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0446\u0435\u043b\u043e\u0435 \u0441\u0435\u043c\u0435\u0439\u0441\u0442\u0432\u043e \u043e\u0446\u0435\u043d\u043e\u043a \u0441\u0432\u0435\u0440\u0445\u0443, \u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0431\u0435\u0440\u0435\u043c \u043b\u0443\u0447\u0448\u0443\u044e. \u041f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\psi^{*}\" alt=\"\\psi^{*}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg\" width=\"20\" height=\"18\" data-width=\"2.461\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.461ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043e\u0442 \u043b\u043e\u0433-\u0441\u0442\u0430\u0442\u0441\u0443\u043c\u043c\u044b \/ \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u044f\u0449\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043c\u043e\u043c\u0435\u043d\u0442\u043e\u0432 <img decoding=\"async\" class=\"formula inline\" source=\"\\psi\" alt=\"\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg\" width=\"12\" height=\"18\" data-width=\"1.473\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.473ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0435\u0439 \u0443\u043a\u043b\u043e\u043d\u0435\u043d\u0438\u044f:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\alpha_{n} \\leq e^{- n\\psi^{*}(\\gamma)},\\ \\ \\psi^{*}(\\gamma) = \\sup_{\\lambda}\\left( \\lambda\\gamma - \\psi(\\lambda) \\right) = KL\\left( P_{\\lambda^{*}(\\gamma)}||P_{0} \\right)\" alt=\"\\alpha_{n} \\leq e^{- n\\psi^{*}(\\gamma)},\\ \\ \\psi^{*}(\\gamma) = \\sup_{\\lambda}\\left( \\lambda\\gamma - \\psi(\\lambda) \\right) = KL\\left( P_{\\lambda^{*}(\\gamma)}||P_{0} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/364\/364103eb92fb451bed1cd2fbf313ad72.svg\" width=\"456\" height=\"34\" data-width=\"57.027\" data-height=\"4.307\" data-vertical-align=\"-1.588\" style=\"width: 57.027ex; height: 4.307ex; vertical-align: -1.588ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/364\/364103eb92fb451bed1cd2fbf313ad72.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/364\/364103eb92fb451bed1cd2fbf313ad72.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0417\u0430\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\psi\" alt=\"\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg\" width=\"12\" height=\"18\" data-width=\"1.473\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.473ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u0442\u043e\u0442 \u0436\u0435 logsumexp \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0449\u0438\u043a f \u0438\u0437 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043a\u043b\u0430\u0441\u0441\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"P_{\\lambda}(x) = \\frac{P_{0}(x)^{1 - \\lambda}P_{1}(x)^{\\lambda}}{e^{\\psi(\\lambda)}} = P_{0}(x)e^{\\lambda s(x) - \\psi(\\lambda)}\" alt=\"P_{\\lambda}(x) = \\frac{P_{0}(x)^{1 - \\lambda}P_{1}(x)^{\\lambda}}{e^{\\psi(\\lambda)}} = P_{0}(x)e^{\\lambda s(x) - \\psi(\\lambda)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c59\/c595cd4a7aca2f71702bab8194689c47.svg\" width=\"344\" height=\"43\" data-width=\"42.974\" data-height=\"5.326\" data-vertical-align=\"-2.097\" style=\"width: 42.974ex; height: 5.326ex; vertical-align: -2.097ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c59\/c595cd4a7aca2f71702bab8194689c47.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c59\/c595cd4a7aca2f71702bab8194689c47.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u0443\u0436\u0435 \u0440\u0430\u0431\u043e\u0442\u0430\u043b\u0438 \u0441 \u043f\u043e\u0434\u043e\u0431\u043d\u043e\u0439 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u0435\u0439 \u0432 \u0440\u0430\u0437\u0434\u0435\u043b\u0435 \u0421\u0442\u0430\u0442\u0444\u0438\u0437: <img decoding=\"async\" class=\"formula inline\" source=\"- s\" alt=\"- s\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba3\/ba37160ce1b14e34a6f860c444bd019d.svg\" width=\"23\" height=\"12\" data-width=\"2.821\" data-height=\"1.505\" data-vertical-align=\"-0.186\" style=\"width: 2.821ex; height: 1.505ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba3\/ba37160ce1b14e34a6f860c444bd019d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba3\/ba37160ce1b14e34a6f860c444bd019d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u043d\u0435\u0440\u0433\u0438\u044f, <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda\" alt=\"\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg\" width=\"12\" height=\"18\" data-width=\"1.319\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.319ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0430, <img decoding=\"async\" class=\"formula inline\" source=\"\\psi\" alt=\"\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg\" width=\"12\" height=\"18\" data-width=\"1.473\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.473ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a11\/a11bd56a0ff5973a5604bb3fc9142b1d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u0430\u044f \u044d\u043d\u0435\u0440\u0433\u0438\u044f, <img decoding=\"async\" class=\"formula inline\" source=\"\\psi^{*}\" alt=\"\\psi^{*}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg\" width=\"20\" height=\"18\" data-width=\"2.461\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.461ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c5\/c57\/c571cd03b75e14be6c6a91a71887249c.svg 781w\" loading=\"lazy\" decode=\"async\"\/> &#8212; \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f, <img decoding=\"async\" class=\"formula inline\" source=\"P_{\\lambda}\" alt=\"P_{\\lambda}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/10e\/10e96cdbf2e1d9339adc3deb8af0205a.svg\" width=\"21\" height=\"18\" data-width=\"2.573\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.573ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/10e\/10e96cdbf2e1d9339adc3deb8af0205a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/10e\/10e96cdbf2e1d9339adc3deb8af0205a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0413\u0438\u0431\u0431\u0441\u0430 (\u0441\u043e\u0444\u0442\u043c\u0430\u043a\u0441 \u0441 \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{1}{\\lambda}\" alt=\"\\frac{1}{\\lambda}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/ca6\/ca6f4a4ef6ffeb8a65665d5a3cdc2187.svg\" width=\"19\" height=\"37\" data-width=\"2.314\" data-height=\"4.615\" data-vertical-align=\"-1.742\" style=\"width: 2.314ex; height: 4.615ex; vertical-align: -1.742ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/ca6\/ca6f4a4ef6ffeb8a65665d5a3cdc2187.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/ca6\/ca6f4a4ef6ffeb8a65665d5a3cdc2187.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u041d\u0430\u043a\u043b\u043e\u043d <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda^{*}\" alt=\"\\lambda^{*}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a19\/a193e83d8a2fa02ddceef33047bdbd66.svg\" width=\"18\" height=\"18\" data-width=\"2.307\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.307ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a19\/a193e83d8a2fa02ddceef33047bdbd66.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a1\/a19\/a193e83d8a2fa02ddceef33047bdbd66.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438\u0449\u0435\u043c \u0442\u0430\u043a, \u0447\u0442\u043e\u0431\u044b \u0440\u0435\u0434\u043a\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\gamma\" alt=\"\\gamma\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/ae5\/ae539dfcc999c28e25a0f3ae65c1de79.svg\" width=\"12\" height=\"18\" data-width=\"1.229\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.229ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/ae5\/ae539dfcc999c28e25a0f3ae65c1de79.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ae\/ae5\/ae539dfcc999c28e25a0f3ae65c1de79.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441\u0442\u0430\u043b\u043e \u0442\u0438\u043f\u0438\u0447\u043d\u044b\u043c \u0441\u0440\u0435\u0434\u043d\u0438\u043c, \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u043c \u043c\u0435\u0442\u043e\u0434 \u043f\u0435\u0440\u0435\u0432\u0430\u043b\u0430.<\/p>\n<p>\u0412\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0434\u0430\u043d\u043d\u043e\u0435 \u0438\u043b\u0438 \u0431\u043e\u043b\u0435\u0435 \u044d\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u043f\u0440\u0438 \u0432\u0435\u0440\u043d\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"H_{0}\" alt=\"H_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg\" width=\"23\" height=\"15\" data-width=\"2.868\" data-height=\"1.92\" data-vertical-align=\"-0.375\" style=\"width: 2.868ex; height: 1.92ex; vertical-align: -0.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430\u0437\u044b\u0432\u0430\u044e\u0442 p-value. \u0415\u0433\u043e \u0441\u0447\u0438\u0442\u0430\u044e\u0442 \u043f\u043e \u0434\u0430\u043d\u043d\u044b\u043c, \u0438 \u0435\u0441\u043b\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"pvalue \\leq \\alpha_{0}\" alt=\"pvalue \\leq \\alpha_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cb9\/cb9ef954e32237ba7f9373e4765292ac.svg\" width=\"95\" height=\"18\" data-width=\"11.907\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 11.907ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cb9\/cb9ef954e32237ba7f9373e4765292ac.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cb9\/cb9ef954e32237ba7f9373e4765292ac.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442\u043e \u043e\u0442\u043a\u043b\u043e\u043d\u0435\u043d\u0438\u0435 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u0441\u0447\u0438\u0442\u0430\u044e\u0442 \u043d\u0435\u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u043d\u043e \u0431\u043e\u043b\u044c\u0448\u0438\u043c \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"H_{0}\" alt=\"H_{0}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg\" width=\"23\" height=\"15\" data-width=\"2.868\" data-height=\"1.92\" data-vertical-align=\"-0.375\" style=\"width: 2.868ex; height: 1.92ex; vertical-align: -0.375ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c5\/7c5081abe6c2100f0e44396b6ac51661.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043e\u0442\u0432\u0435\u0440\u0433\u0430\u044e\u0442. \u041e\u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c, \u0447\u0442\u043e:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"pvalue = P_{0}\\left( S_{n} \\geq \\gamma \\right)\\\u00a0 \\propto \\ e^{- n\\psi^{*}(\\gamma)},\\ \\ KL\\left( {\\widehat{P}}_{n}||P_{0} \\right) \\approx - \\frac{1}{n}\\ln(pvalue) \\rightarrow \\psi^{*}(\\gamma)\" alt=\"pvalue = P_{0}\\left( S_{n} \\geq \\gamma \\right)\\\u00a0 \\propto \\ e^{- n\\psi^{*}(\\gamma)},\\ \\ KL\\left( {\\widehat{P}}_{n}||P_{0} \\right) \\approx - \\frac{1}{n}\\ln(pvalue) \\rightarrow \\psi^{*}(\\gamma)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a4\/3a4517b06c0579f54d1ff339d0e6ea8c.svg\" width=\"588\" height=\"37\" data-width=\"73.544\" data-height=\"4.613\" data-vertical-align=\"-1.741\" style=\"width: 73.544ex; height: 4.613ex; vertical-align: -1.741ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a4\/3a4517b06c0579f54d1ff339d0e6ea8c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3a\/3a4\/3a4517b06c0579f54d1ff339d0e6ea8c.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u0438\u043d\u0443\u0441 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c p-value \u2013 \u044d\u0442\u043e \u0430\u043d\u0430\u043b\u043e\u0433 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 \u0438 \u043c\u0435\u0440\u0430 \u043d\u0430\u0448\u0435\u0433\u043e \u0443\u0434\u0438\u0432\u043b\u0435\u043d\u0438\u044f \u0434\u0430\u043d\u043d\u044b\u043c (\u043d\u0430\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0441\u0438\u043b\u044c\u043d\u043e \u043e\u0442\u043b\u0438\u0447\u0430\u044e\u0442\u0441\u044f \u044d\u043c\u043f\u0438\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0438 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u043c\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f). \u0421\u0430\u043c\u043e p-value \u2013 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u0445\u0432\u043e\u0441\u0442\u0430 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f: \u043a\u043e\u0433\u0434\u0430 \u0443 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u0435\u0441\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u044f\u0449\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f, \u0435\u0451 \u0445\u0432\u043e\u0441\u0442 \u0443\u043f\u0440\u0430\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432\u044b\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c \u044d\u0442\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/p>\n<p>\u0418\u0442\u043e\u0433: \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u0440\u043e\u043d\u0438\u0437\u044b\u0432\u0430\u0435\u0442 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0443, \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044f \u043c\u043d\u043e\u0433\u043e \u0446\u0435\u043d\u043d\u044b\u0445 \u0438\u0434\u0435\u0439 \u0438 \u043c\u0435\u0442\u043e\u0434\u043e\u0432<\/p>\n<hr\/>\n<h2>ML<\/h2>\n<p>\u0411\u0430\u0439\u0435\u0441\u043e\u0432\u0441\u043a\u0438\u0439 \u0432\u044b\u0432\u043e\u0434<\/p>\n<p>\u041a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u0440\u0430\u0431\u043e\u0442\u0430 \u0411\u0430\u0439\u0435\u0441\u0430 \u0431\u044b\u043b\u0430 \u043e\u043f\u0443\u0431\u043b\u0438\u043a\u043e\u0432\u0430\u043d\u0430 \u043f\u043e\u0441\u043c\u0435\u0440\u0442\u043d\u043e \u2013 \u0438\u043c\u0435\u043d\u043d\u043e \u043e\u043d\u0430 \u0437\u0430\u043b\u043e\u0436\u0438\u043b\u0430 \u043e\u0441\u043d\u043e\u0432\u044b \u0431\u0430\u0439\u0435\u0441\u043e\u0432\u0441\u043a\u043e\u0439 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u0438 \u0437\u043d\u0430\u0447\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0447\u0430\u0441\u0442\u0438 \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0433\u043e ML. \u0411\u0430\u0439\u0435\u0441 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043b \u0442\u0430\u043a\u043e\u0439 \u043c\u044b\u0441\u043b\u0435\u043d\u043d\u044b\u0439 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442:<\/p>\n<ol>\n<li>\n<p>\u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u0442\u043e\u0440 \u0441 \u0437\u0430\u0432\u044f\u0437\u0430\u043d\u043d\u044b\u043c\u0438 \u0433\u043b\u0430\u0437\u0430\u043c\u0438 \u043a\u0438\u0434\u0430\u0435\u0442 \u0431\u0435\u043b\u044b\u0439 \u0448\u0430\u0440 \u043d\u0430 \u0431\u0438\u043b\u044c\u044f\u0440\u0434\u043d\u044b\u0439 \u0441\u0442\u043e\u043b \u0438 \u0445\u043e\u0447\u0435\u0442 \u0443\u0437\u043d\u0430\u0442\u044c, \u043a\u0443\u0434\u0430 \u043e\u043d \u043f\u043e\u043f\u0430\u043b (<img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u043e\u0437\u0438\u0446\u0438\u044f \u0448\u0430\u0440\u0430 \u043d\u0430 \u0441\u0442\u043e\u043b\u0435, 0 \u2013 \u043b\u0435\u0432\u044b\u0439 \u043a\u0440\u0430\u0439, 1 &#8212; \u043f\u0440\u0430\u0432\u044b\u0439)<\/p>\n<\/li>\n<li>\n<p>\u0434\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u0442\u043e\u0440 \u043a\u0438\u0434\u0430\u0435\u0442 \u043d\u0430 \u0441\u0442\u043e\u043b \u0434\u0440\u0443\u0433\u043e\u0439 \u0448\u0430\u0440, \u0430 \u0430\u0441\u0441\u0438\u0441\u0442\u0435\u043d\u0442 \u043f\u043e\u0434\u0441\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442, \u043a\u0443\u0434\u0430 \u0442\u043e\u0442 \u043f\u0440\u0438\u0437\u0435\u043c\u043b\u0438\u043b\u0441\u044f \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0431\u0435\u043b\u043e\u0433\u043e. \u042d\u0442\u0430 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u043f\u043e\u043c\u043e\u0433\u0430\u0435\u0442 \u0441\u043a\u043e\u0440\u0440\u0435\u043a\u0442\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043e \u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0438 \u0431\u0435\u043b\u043e\u0433\u043e \u0448\u0430\u0440\u0430<\/p>\n<\/li>\n<li>\n<p>\u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u0430 \u043f\u043e\u0432\u0442\u043e\u0440\u044f\u0435\u0442\u0441\u044f n \u0440\u0430\u0437, \u043f\u043e\u043a\u0430 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u0442\u043e\u0440 \u043d\u0435 \u0434\u043e\u0441\u0442\u0438\u0433\u043d\u0435\u0442 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e\u0439 \u0443\u0432\u0435\u0440\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0432 \u043e\u0442\u0432\u0435\u0442\u0435<\/p>\n<\/li>\n<\/ol>\n<p><img decoding=\"async\" class=\"formula\" source=\"P\\left( \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta\\ |\\ \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445 \\right) = \\frac{P\\left( \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445\\ |\\ \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta \\right) \\cdot P(\u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta)}{P(\u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445)}\" alt=\"P\\left( \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta\\ |\\ \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445 \\right) = \\frac{P\\left( \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445\\ |\\ \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta \\right) \\cdot P(\u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta)}{P(\u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ec\/ec7\/ec72c3c5cd60196eec6f6383085ab31e.svg\" width=\"762\" height=\"44\" data-width=\"95.297\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 95.297ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ec\/ec7\/ec72c3c5cd60196eec6f6383085ab31e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ec\/ec7\/ec72c3c5cd60196eec6f6383085ab31e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"P\\left( \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta\\ |\\ \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445 \\right)\" alt=\"P\\left( \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta\\ |\\ \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445 \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fc\/fc8\/fc804bda2bb1cfaa332fc5e10e1daebb.svg\" width=\"298\" height=\"18\" data-width=\"37.248\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 37.248ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fc\/fc8\/fc804bda2bb1cfaa332fc5e10e1daebb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fc\/fc8\/fc804bda2bb1cfaa332fc5e10e1daebb.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440 (\u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0435, \u0443\u0442\u043e\u0447\u043d\u0435\u043d\u043d\u043e\u0435 \u043f\u043e \u0434\u0430\u043d\u043d\u044b\u043c)<\/p>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"P(\u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta)\" alt=\"P(\u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/922\/9224767d6cd55b67572c3cdb954a3791.svg\" width=\"121\" height=\"18\" data-width=\"15.154\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 15.154ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/922\/9224767d6cd55b67572c3cdb954a3791.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/922\/9224767d6cd55b67572c3cdb954a3791.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0430\u043f\u0440\u0438\u043e\u0440 (\u0442\u0435\u043a\u0443\u0449\u0430\u044f \u0433\u0438\u043f\u043e\u0442\u0435\u0437\u0430), \u0432 \u043d\u0430\u0447\u0430\u043b\u0435 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u043c, \u0447\u0442\u043e \u0448\u0430\u0440 \u043f\u0440\u0438\u0437\u0435\u043c\u043b\u044f\u0435\u0442\u0441\u044f \u0440\u0430\u0432\u043d\u043e\u043c\u0435\u0440\u043d\u043e, \u0442.\u0435. \u0438\u043c\u0435\u0435\u0442 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"Beta(1,1) \\equiv 1\" alt=\"Beta(1,1) \\equiv 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/35\/355\/3552c3bfdba637e322567d3d5b109438.svg\" width=\"112\" height=\"18\" data-width=\"13.962\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 13.962ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/35\/355\/3552c3bfdba637e322567d3d5b109438.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/35\/355\/3552c3bfdba637e322567d3d5b109438.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"P\\left( \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445\\ |\\ \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta \\right)\" alt=\"P\\left( \u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445\\ |\\ \u0431\u0435\u043b\u044b\u0439\\ \u043d\u0430\\ \\theta \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/dee\/deefef5cad3c62e5b4e6a115262dddfb.svg\" width=\"298\" height=\"18\" data-width=\"37.248\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 37.248ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/dee\/deefef5cad3c62e5b4e6a115262dddfb.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/de\/dee\/deefef5cad3c62e5b4e6a115262dddfb.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0437\u043d\u0430\u043a\u043e\u043c\u043e\u0435 \u043d\u0430\u043c \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u0435, \u0443 \u043d\u0430\u0441 \u043e\u043d\u043e \u0440\u0430\u0432\u043d\u043e <img decoding=\"async\" class=\"formula inline\" source=\"C_{n}^{k}\\theta^{k}(1 - \\theta)^{n - k}\" alt=\"C_{n}^{k}\\theta^{k}(1 - \\theta)^{n - k}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/90c\/90c91c62312edcca5e30ba2a37b20fd7.svg\" width=\"119\" height=\"21\" data-width=\"14.885\" data-height=\"2.621\" data-vertical-align=\"-0.745\" style=\"width: 14.885ex; height: 2.621ex; vertical-align: -0.745ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/90c\/90c91c62312edcca5e30ba2a37b20fd7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/90\/90c\/90c91c62312edcca5e30ba2a37b20fd7.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u0431\u0438\u043d\u043e\u043c\u0438\u0430\u043b\u044c\u043d\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 (\u0441\u0443\u043c\u043c\u0430 n \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u044b\u0445 \u0438\u0441\u043f\u044b\u0442\u0430\u043d\u0438\u0439 \u0411\u0435\u0440\u043d\u0443\u043b\u043b\u0438)<\/p>\n<p><img decoding=\"async\" class=\"formula inline\" source=\"P(\u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445)\" alt=\"P(\u0441\u043f\u0440\u0430\u0432\u0430\\ k\\ \u0446\u0432\u0435\u0442\u043d\u044b\u0445)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8f\/f8f5509830eccdec261b8a5be0d8eb8e.svg\" width=\"187\" height=\"18\" data-width=\"23.416\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 23.416ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8f\/f8f5509830eccdec261b8a5be0d8eb8e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8f\/f8f5509830eccdec261b8a5be0d8eb8e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f evidence. \u041f\u043e \u0441\u0443\u0442\u0438, \u043d\u0430\u043c \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u0435\u043d \u0442\u043e\u043b\u044c\u043a\u043e \u0447\u0438\u0441\u043b\u0438\u0442\u0435\u043b\u044c, \u0430 \u0437\u043d\u0430\u043c\u0435\u043d\u0430\u0442\u0435\u043b\u044c &#8212; \u044d\u0442\u043e \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043e\u0447\u043d\u0430\u044f \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430, \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u0447\u0430\u0441\u0442\u043e \u043e\u0446\u0435\u043d\u0438\u0432\u0430\u044e\u0442 \u0438\u043b\u0438 \u043f\u043e\u0434\u0433\u043e\u043d\u044f\u044e\u0442<\/p>\n<p>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u0411\u0430\u0439\u0435\u0441\u0430 \u043e\u0431\u044a\u0435\u0434\u0438\u043d\u044f\u0435\u0442 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0439 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 \u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043f\u043e\u043b\u043d\u043e\u0439 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"P\\left( \\theta\\ |\\ k \\right) = \\frac{P(\\theta,k)}{P(k)} = \\frac{P\\left( k\\ |\\ \\theta \\right) \\cdot P(\\theta)}{\\sum_{\\theta}^{}{P\\left( k\\ |\\ \\theta \\right) \\cdot P(\\theta)}}\" alt=\"P\\left( \\theta\\ |\\ k \\right) = \\frac{P(\\theta,k)}{P(k)} = \\frac{P\\left( k\\ |\\ \\theta \\right) \\cdot P(\\theta)}{\\sum_{\\theta}^{}{P\\left( k\\ |\\ \\theta \\right) \\cdot P(\\theta)}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/18\/189\/189e0ff27d31cd5342ba404dccd9d5cf.svg\" width=\"322\" height=\"45\" data-width=\"40.262\" data-height=\"5.571\" data-vertical-align=\"-2.22\" style=\"width: 40.262ex; height: 5.571ex; vertical-align: -2.22ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/18\/189\/189e0ff27d31cd5342ba404dccd9d5cf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/18\/189\/189e0ff27d31cd5342ba404dccd9d5cf.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412 \u043d\u0430\u0448\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"P\\left( \\theta\\ |\\ k \\right) = \\frac{C_{n}^{k}\\theta^{k}(1 - \\theta)^{n - k} \\cdot 1}{(n + 1)^{- 1}}\\\u00a0 \\propto \\ \\theta^{k}(1 - \\theta)^{n - k},\\ \\ \u044d\u0442\u043e\\ Beta(k + 1,n - k + 1)\" alt=\"P\\left( \\theta\\ |\\ k \\right) = \\frac{C_{n}^{k}\\theta^{k}(1 - \\theta)^{n - k} \\cdot 1}{(n + 1)^{- 1}}\\\u00a0 \\propto \\ \\theta^{k}(1 - \\theta)^{n - k},\\ \\ \u044d\u0442\u043e\\ Beta(k + 1,n - k + 1)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6e\/6ed\/6ed838ea9ed31e2a7ef51947133ce1e4.svg\" width=\"594\" height=\"46\" data-width=\"74.276\" data-height=\"5.743\" data-vertical-align=\"-2.306\" style=\"width: 74.276ex; height: 5.743ex; vertical-align: -2.306ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6e\/6ed\/6ed838ea9ed31e2a7ef51947133ce1e4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6e\/6ed\/6ed838ea9ed31e2a7ef51947133ce1e4.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0427\u0442\u043e \u0437\u0434\u0435\u0441\u044c \u043c\u043e\u0436\u043d\u043e \u0437\u0430\u043c\u0435\u0442\u0438\u0442\u044c:<\/p>\n<ol>\n<li>\n<p>\u043c\u043e\u0436\u043d\u043e \u043d\u0435 \u0445\u0440\u0430\u043d\u0438\u0442\u044c \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044e \u043e \u0432\u0441\u0435\u0445 \u0448\u0430\u0440\u0430\u0445, \u0430 \u0442\u043e\u043b\u044c\u043a\u043e \u043e\u0431\u0449\u0435\u0435 \u043a\u043e\u043b-\u0432\u043e \u0448\u0430\u0440\u043e\u0432 \u0438 \u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0438\u0437 \u043d\u0438\u0445 \u0441\u043f\u0440\u0430\u0432\u0430 \u043e\u0442 \u0438\u0441\u043a\u043e\u043c\u043e\u0433\u043e, \u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"(n,k)\" alt=\"(n,k)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/13\/13b\/13be0c14eaff4ce2fa73860c7e5931aa.svg\" width=\"42\" height=\"18\" data-width=\"5.302\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.302ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/13\/13b\/13be0c14eaff4ce2fa73860c7e5931aa.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/13\/13b\/13be0c14eaff4ce2fa73860c7e5931aa.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u0430\u044f \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0430<\/p>\n<\/li>\n<li>\n<p>\u043c\u044b \u043f\u0435\u0440\u0435\u0448\u043b\u0438 \u043e\u0442 \u043e\u0434\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043a \u0434\u0440\u0443\u0433\u043e\u043c\u0443, \u043f\u0440\u0438\u0447\u0435\u043c \u0411\u0435\u0440\u043d\u0443\u043b\u043b\u0438, \u043a\u0430\u043a \u043c\u044b \u0434\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u043b\u0438, \u043b\u0435\u0436\u0438\u0442 \u0432 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u043c \u043a\u043b\u0430\u0441\u0441\u0435 (\u043d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435, \u0431\u0438\u043d\u043e\u043c\u0438\u0430\u043b\u044c\u043d\u043e\u0435 \u0438 \u0431\u0435\u0442\u0430 \u2013 \u0442\u043e\u0436\u0435)<\/p>\n<\/li>\n<li>\n<p>\u0438\u0442\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u0430\u044f \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u0430 \u0443\u0442\u043e\u0447\u043d\u0435\u043d\u0438\u044f \u0438\u0434\u0435\u0439\u043d\u043e \u043d\u0430\u043f\u043e\u043c\u0438\u043d\u0430\u0435\u0442 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u043f\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f\u043c (\u0432\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043a \u044d\u0442\u043e\u043c\u0443 \u043f\u043e\u0437\u0436\u0435)<\/p>\n<\/li>\n<li>\n<p>\u0443\u0442\u043e\u0447\u043d\u0435\u043d\u0438\u0435 \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u043a\u0430\u043a \u0434\u043e\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u0431\u0438\u043d\u043e\u043c\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f \u043d\u0430 \u0431\u0435\u0442\u0430-\u0430\u043f\u0440\u0438\u043e\u0440. \u041f\u0440\u0438 \u044d\u0442\u043e\u043c \u0441\u0430\u043c \u0432\u0438\u0434 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f \u043d\u0435 \u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<\/li>\n<\/ol>\n<p>\u0415\u0441\u043b\u0438 \u0430\u043f\u0440\u0438\u043e\u0440 \u043d\u0435 \u043c\u0435\u043d\u044f\u0435\u0442 \u0432\u0438\u0434 \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440\u0430, \u0442\u043e \u0430\u043f\u0440\u0438\u043e\u0440 \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u044b\u043c \u043a \u043d\u0435\u043c\u0443. \u0421\u043b\u043e\u0432\u043e \u00ab\u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u044b\u0439\u00bb \u043d\u0430\u043c\u0435\u043a\u0430\u0435\u0442 \u043d\u0430 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430. \u041f\u0443\u0441\u0442\u044c \u0434\u0430\u043d\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438\u0437 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043a\u043b\u0430\u0441\u0441\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p\\left( x \\middle| \\theta \\right) = h(x)\\exp{\\left( \\theta T(x) - f(\\theta) \\right),\\ \\ \u0433\u0434\u0435\\ f(\\theta) = \\ln{\\int_{}^{}{h(x)\\exp{\\left( \\theta T(x) \\right)dx}}}}\" alt=\"p\\left( x \\middle| \\theta \\right) = h(x)\\exp{\\left( \\theta T(x) - f(\\theta) \\right),\\ \\ \u0433\u0434\u0435\\ f(\\theta) = \\ln{\\int_{}^{}{h(x)\\exp{\\left( \\theta T(x) \\right)dx}}}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/71\/71a\/71ae3c2d498b604c6df506d517c3ad6f.svg\" width=\"557\" height=\"41\" data-width=\"69.632\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 69.632ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/71\/71a\/71ae3c2d498b604c6df506d517c3ad6f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/71\/71a\/71ae3c2d498b604c6df506d517c3ad6f.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u0435 \u0434\u043b\u044f \u0432\u0441\u0435 \u0432\u044b\u0431\u043e\u0440\u043a\u0438 \u0440\u0430\u0437\u043c\u0435\u0440\u0430 n:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p\\left( X \\middle| \\theta \\right) = \\prod_{}^{}{h\\left( x_{i} \\right)} \\cdot \\exp\\left( \\theta\\sum_{}^{}{T\\left( x_{i} \\right)} - nf(\\theta) \\right)\" alt=\"p\\left( X \\middle| \\theta \\right) = \\prod_{}^{}{h\\left( x_{i} \\right)} \\cdot \\exp\\left( \\theta\\sum_{}^{}{T\\left( x_{i} \\right)} - nf(\\theta) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5e\/5e5\/5e5fd6d5f89fe126002a11fc75dd2958.svg\" width=\"374\" height=\"54\" data-width=\"46.74\" data-height=\"6.785\" data-vertical-align=\"-2.827\" style=\"width: 46.74ex; height: 6.785ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5e\/5e5\/5e5fd6d5f89fe126002a11fc75dd2958.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5e\/5e5\/5e5fd6d5f89fe126002a11fc75dd2958.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0427\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u044b\u0439 \u0430\u043f\u0440\u0438\u043e\u0440, \u043c\u044b \u0442\u0440\u0435\u0431\u0443\u0435\u043c, \u0447\u0442\u043e\u0431\u044b \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440\u043d\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u043e\u0445\u0440\u0430\u043d\u0438\u043b\u043e \u0444\u043e\u0440\u043c\u0443. \u0415\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0439 \u0441\u043f\u043e\u0441\u043e\u0431 \u044d\u0442\u043e \u0441\u0434\u0435\u043b\u0430\u0442\u044c &#8212; \u0437\u0430\u0434\u0430\u0442\u044c \u0430\u043f\u0440\u0438\u043e\u0440 \u0432 \u0432\u0438\u0434\u0435:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p\\left( \\theta \\middle| \\alpha,\\nu \\right) = \\exp\\left( \\theta\\alpha - \\nu f(\\theta) - g(\\alpha,\\nu) \\right)\" alt=\"p\\left( \\theta \\middle| \\alpha,\\nu \\right) = \\exp\\left( \\theta\\alpha - \\nu f(\\theta) - g(\\alpha,\\nu) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6f\/6ff\/6ffa8ac7e2c33b04bd6d9381c470a4f4.svg\" width=\"296\" height=\"18\" data-width=\"37.027\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 37.027ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6f\/6ff\/6ffa8ac7e2c33b04bd6d9381c470a4f4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6f\/6ff\/6ffa8ac7e2c33b04bd6d9381c470a4f4.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0413\u0434\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\alpha\" alt=\"\\alpha\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg\" width=\"12\" height=\"18\" data-width=\"1.448\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.448ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0441\u0443\u043c\u043c\u0430 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u044b\u0445 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a \u0438\u0437 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0445 \u043d\u0430 \u0434\u0430\u043d\u043d\u044b\u0439 \u043c\u043e\u043c\u0435\u043d\u0442 (\u0438\u043b\u0438 \u0433\u0438\u043f\u043e\u0442\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445) \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u0439, <img decoding=\"async\" class=\"formula inline\" source=\"\\nu\" alt=\"\\nu\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fd\/4fdefba26320686bb2bd0579a0df421c.svg\" width=\"12\" height=\"18\" data-width=\"1.199\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.199ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fd\/4fdefba26320686bb2bd0579a0df421c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4fd\/4fdefba26320686bb2bd0579a0df421c.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0438\u0445 \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e (\u0432 \u0433\u0438\u043f\u043e\u0442\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u2013 \u043c\u0435\u0440\u0430 \u0443\u0432\u0435\u0440\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0432 \u0430\u043f\u0440\u0438\u043e\u0440\u0435), <img decoding=\"async\" class=\"formula inline\" source=\"g(\\alpha,\\nu)\" alt=\"g(\\alpha,\\nu)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea8\/ea8ed8dee0600c173495ba189940ad37.svg\" width=\"52\" height=\"18\" data-width=\"6.492\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.492ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea8\/ea8ed8dee0600c173495ba189940ad37.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea8\/ea8ed8dee0600c173495ba189940ad37.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043e\u0447\u043d\u0430\u044f \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430. \u041c\u044b \u0445\u043e\u0442\u0438\u043c \u0442\u0430\u043a \u043f\u043e\u0434\u043e\u0431\u0440\u0430\u0442\u044c \u044d\u0442\u0443 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0443, \u0447\u0442\u043e\u0431\u044b \u0430\u043f\u0440\u0438\u043e\u0440 \u0431\u044b\u043b \u043a\u043e\u0440\u0440\u0435\u043a\u0442\u043d\u044b\u043c, \u0442.\u0435. \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u043e\u0432\u0430\u043b\u0441\u044f \u0432 \u0435\u0434\u0438\u043d\u0438\u0446\u0443:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"g(\\alpha,\\nu) = \\ln{\\int_{}^{}{\\exp\\left( \\theta\\alpha - \\nu f(\\theta) \\right)d\\theta}}\" alt=\"g(\\alpha,\\nu) = \\ln{\\int_{}^{}{\\exp\\left( \\theta\\alpha - \\nu f(\\theta) \\right)d\\theta}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3d7\/3d7e8a9043df32431fd67853ee670ba3.svg\" width=\"264\" height=\"41\" data-width=\"33.002\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 33.002ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3d7\/3d7e8a9043df32431fd67853ee670ba3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3d\/3d7\/3d7e8a9043df32431fd67853ee670ba3.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0420\u0435\u0448\u0430\u0435\u0442\u0441\u044f \u0443\u0436\u0435 \u0437\u043d\u0430\u043a\u043e\u043c\u044b\u043c \u043d\u0430\u043c \u043c\u0435\u0442\u043e\u0434\u043e\u043c \u043f\u0435\u0440\u0435\u0432\u0430\u043b\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"g(\\alpha,\\nu) \\approx \\sup_{\\theta}\\left( \\theta\\alpha - \\nu f(\\theta) \\right) = \\nu \\cdot \\sup_{\\theta}\\left( \\theta \\cdot \\frac{\\alpha}{\\nu} - f(\\theta) \\right)\" alt=\"g(\\alpha,\\nu) \\approx \\sup_{\\theta}\\left( \\theta\\alpha - \\nu f(\\theta) \\right) = \\nu \\cdot \\sup_{\\theta}\\left( \\theta \\cdot \\frac{\\alpha}{\\nu} - f(\\theta) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/58\/58d\/58d7ac35e29a40a5f2738e1fbfef7eb5.svg\" width=\"399\" height=\"38\" data-width=\"49.9\" data-height=\"4.788\" data-vertical-align=\"-1.828\" style=\"width: 49.9ex; height: 4.788ex; vertical-align: -1.828ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/58\/58d\/58d7ac35e29a40a5f2738e1fbfef7eb5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/58\/58d\/58d7ac35e29a40a5f2738e1fbfef7eb5.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c, \u0447\u0442\u043e\u0431\u044b \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0430\u043f\u0440\u0438\u043e\u0440, \u0442\u0440\u0435\u0431\u0443\u0435\u0442\u0441\u044f \u0432\u0437\u044f\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"g\" alt=\"g\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b2\/b2f\/b2f5ff47436671b6e533d8dc3614845d.svg\" width=\"12\" height=\"12\" data-width=\"1.079\" data-height=\"1.464\" data-vertical-align=\"-0.464\" style=\"width: 1.079ex; height: 1.464ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b2\/b2f\/b2f5ff47436671b6e533d8dc3614845d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b2\/b2f\/b2f5ff47436671b6e533d8dc3614845d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043a\u0430\u043a \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043e\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"f\" alt=\"f\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8f\/8fa\/8fa14cdd754f91cc6554c9e71929cce7.svg\" width=\"12\" height=\"16\" data-width=\"1.244\" data-height=\"2.059\" data-vertical-align=\"-0.464\" style=\"width: 1.244ex; height: 2.059ex; vertical-align: -0.464ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8f\/8fa\/8fa14cdd754f91cc6554c9e71929cce7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8f\/8fa\/8fa14cdd754f91cc6554c9e71929cce7.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"g(\\alpha,\\nu) = \\nu \\cdot f^{*}\\left( \\frac{\\alpha}{\\nu} \\right)\" alt=\"g(\\alpha,\\nu) = \\nu \\cdot f^{*}\\left( \\frac{\\alpha}{\\nu} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cd\/cd9\/cd95a95ac265a0fbc3a8f6d6139339f7.svg\" width=\"162\" height=\"33\" data-width=\"20.217\" data-height=\"4.153\" data-vertical-align=\"-1.511\" style=\"width: 20.217ex; height: 4.153ex; vertical-align: -1.511ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cd\/cd9\/cd95a95ac265a0fbc3a8f6d6139339f7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cd\/cd9\/cd95a95ac265a0fbc3a8f6d6139339f7.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0420\u0435\u0433\u0443\u043b\u044f\u0440\u0438\u0437\u0430\u0446\u0438\u044f \u0432 \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439 \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u0438<\/p>\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0437\u0430\u0434\u0430\u0447\u0443: \u0434\u0430\u043d\u044b \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u0435 \u043c\u0435\u0442\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"Y\" alt=\"Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg\" width=\"14\" height=\"12\" data-width=\"1.726\" data-height=\"1.545\" style=\"width: 1.726ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"X\" alt=\"X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg\" width=\"15\" height=\"12\" data-width=\"1.928\" data-height=\"1.545\" style=\"width: 1.928ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u043c\u0435\u0442\u043a\u0438 \u043f\u043e\u0440\u043e\u0436\u0434\u0435\u043d\u044b \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439 \u043c\u043e\u0434\u0435\u043b\u044c\u044e \u0441 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u044b\u043c \u0448\u0443\u043c\u043e\u043c \u043a\u0430\u043a <img decoding=\"async\" class=\"formula inline\" source=\"y_{i}\\ \\sim\\ N\\left( \\theta^{T}x_{i},\\sigma^{2} \\right)\" alt=\"y_{i}\\ \\sim\\ N\\left( \\theta^{T}x_{i},\\sigma^{2} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/45f\/45fd00b9c58428beb5e72a0de8f7f51e.svg\" width=\"145\" height=\"22\" data-width=\"18.15\" data-height=\"2.808\" data-vertical-align=\"-0.838\" style=\"width: 18.15ex; height: 2.808ex; vertical-align: -0.838ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/45f\/45fd00b9c58428beb5e72a0de8f7f51e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/45f\/45fd00b9c58428beb5e72a0de8f7f51e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0438 \u043f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u043f\u043e\u0434\u043e\u0431\u0440\u0430\u0442\u044c \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0435 \u0432\u0435\u0441\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta}\" alt=\"\\widehat{\\theta}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg\" width=\"12\" height=\"24\" data-width=\"1.061\" data-height=\"2.993\" data-vertical-align=\"-0.931\" style=\"width: 1.061ex; height: 2.993ex; vertical-align: -0.931ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043c\u043e\u0434\u0435\u043b\u0438. \u0421\u0440\u0430\u0437\u0443 \u043e\u0442\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \u0447\u0430\u0441\u0442\u043e \u043c\u043e\u0434\u0435\u043b\u044c \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0442 \u043a\u0430\u043a <img decoding=\"async\" class=\"formula inline\" source=\"y_{i} = \\theta^{T}x_{i} + b_{i}\" alt=\"y_{i} = \\theta^{T}x_{i} + b_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/12\/128\/1284b7d3359a9df67903adc100f7c72e.svg\" width=\"110\" height=\"21\" data-width=\"13.75\" data-height=\"2.583\" data-vertical-align=\"-0.726\" style=\"width: 13.75ex; height: 2.583ex; vertical-align: -0.726ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/12\/128\/1284b7d3359a9df67903adc100f7c72e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/12\/128\/1284b7d3359a9df67903adc100f7c72e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043d\u043e \u044d\u0442\u043e \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e \u0434\u043e\u0431\u0430\u0432\u043b\u0435\u043d\u0438\u044e \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0430 \u0438\u0437 \u0432\u0441\u0435\u0445 \u0435\u0434\u0438\u043d\u0438\u0446 \u0438 \u0432\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"b_{i}\" alt=\"b_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c9f\/c9f6d8557ce40f989fa727b5c0bb1ddf.svg\" width=\"14\" height=\"15\" data-width=\"1.71\" data-height=\"1.927\" data-vertical-align=\"-0.357\" style=\"width: 1.71ex; height: 1.927ex; vertical-align: -0.357ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c9f\/c9f6d8557ce40f989fa727b5c0bb1ddf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c9\/c9f\/c9f6d8557ce40f989fa727b5c0bb1ddf.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432 \u0432\u0435\u043a\u0442\u043e\u0440 \u0432\u0435\u0441\u043e\u0432 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0442\u0430\u043a \u0447\u0442\u043e \u0431\u0443\u0434\u0435\u043c \u043f\u0438\u0441\u0430\u0442\u044c \u0432 \u043f\u0440\u043e\u0441\u0442\u043e\u043c \u0432\u0438\u0434\u0435<\/p>\n<p>\u0427\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442 \u00ab\u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0435\u00bb? \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u043a\u0430\u043a \u041e\u041c\u041f. \u0410\u043b\u044c\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432\u043d\u043e &#8212; \u043c\u043e\u0436\u043d\u043e \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0441\u0440\u0435\u0434\u043d\u044e\u044e \u043e\u0448\u0438\u0431\u043a\u0443 \u043c\u043e\u0434\u0435\u043b\u0438, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, MSE &#8212; \u0441\u0440\u0435\u0434\u043d\u0435\u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u0443\u044e:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"MSE(\\theta) = \\frac{1}{2}\\left\\| Y - X\\theta \\right\\|_{2}^{2} = \\frac{1}{2}(Y - X\\theta)^{T}(Y - X\\theta) = \\frac{1}{2}\\sum_{i = 1}^{n}\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2}\" alt=\"MSE(\\theta) = \\frac{1}{2}\\left\\| Y - X\\theta \\right\\|_{2}^{2} = \\frac{1}{2}(Y - X\\theta)^{T}(Y - X\\theta) = \\frac{1}{2}\\sum_{i = 1}^{n}\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d8e\/d8e6a665d2c348505b9d23d5657340b4.svg\" width=\"564\" height=\"51\" data-width=\"70.449\" data-height=\"6.354\" data-vertical-align=\"-2.611\" style=\"width: 70.449ex; height: 6.354ex; vertical-align: -2.611ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d8e\/d8e6a665d2c348505b9d23d5657340b4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d8e\/d8e6a665d2c348505b9d23d5657340b4.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0418\u0442\u043e\u0433: \u043e\u043f\u0442\u0438\u043c\u0443\u043c \u043c\u043e\u0436\u043d\u043e \u043d\u0430\u0439\u0442\u0438 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438, \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u043c \u0441\u043f\u0443\u0441\u043a\u043e\u043c \u0438 \u043a\u0430\u043a \u041e\u041c\u041f. \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0432\u0441\u0435 \u0432\u0430\u0440\u0438\u0430\u043d\u0442\u044b<\/p>\n<p>1) \u0410\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043f\u043e\u0434\u0445\u043e\u0434 (\u043c\u0435\u0442\u043e\u0434 \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0438\u0445 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432, \u041c\u041d\u041a):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\frac{d}{d\\theta}MSE = X^{T}X\\theta - X^{T}Y = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" alt=\"\\frac{d}{d\\theta}MSE = X^{T}X\\theta - X^{T}Y = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6c\/6cc\/6cc160764d1cb14e7bd18c7747d8200d.svg\" width=\"438\" height=\"37\" data-width=\"54.693\" data-height=\"4.674\" data-vertical-align=\"-1.771\" style=\"width: 54.693ex; height: 4.674ex; vertical-align: -1.771ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6c\/6cc\/6cc160764d1cb14e7bd18c7747d8200d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6c\/6cc\/6cc160764d1cb14e7bd18c7747d8200d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430 \u043f\u0440\u0438 \u043e\u0431\u0440\u0430\u0449\u0435\u043d\u0438\u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u044b <img decoding=\"async\" class=\"formula inline\" source=\"X^{T}X\" alt=\"X^{T}X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d5\/5d5eaf3c0a410adace4935eb816a70a3.svg\" width=\"42\" height=\"16\" data-width=\"5.285\" data-height=\"2.017\" style=\"width: 5.285ex; height: 2.017ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d5\/5d5eaf3c0a410adace4935eb816a70a3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d5\/5d5eaf3c0a410adace4935eb816a70a3.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0415\u0441\u043b\u0438 \u043e\u043d\u0430 \u0432\u044b\u0440\u043e\u0436\u0434\u0435\u043d\u0430, \u043e\u0431\u0440\u0430\u0442\u0438\u0442\u044c \u043d\u0435\u043b\u044c\u0437\u044f \u0432\u043e\u0432\u0441\u0435. \u0415\u0441\u043b\u0438 \u043e\u043d\u0430 \u0431\u043b\u0438\u0437\u043a\u0430 \u043a \u0432\u044b\u0440\u043e\u0436\u0434\u0435\u043d\u043d\u043e\u0439 \u2013 \u043e\u0431\u0440\u0430\u0442\u0438\u0442\u044c \u043c\u043e\u0436\u043d\u043e, \u043d\u043e \u0441\u0430\u043c\u0438 \u0432\u0435\u0441\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441\u0442\u0430\u043d\u043e\u0432\u044f\u0442\u0441\u044f \u043e\u0433\u0440\u043e\u043c\u043d\u044b\u043c\u0438, \u0438 \u043c\u043e\u0434\u0435\u043b\u044c \u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0447\u0435\u0440\u0435\u0441\u0447\u0443\u0440 \u0447\u0443\u0432\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u0430 \u043a\u043e \u0432\u0445\u043e\u0434\u043d\u044b\u043c \u0434\u0430\u043d\u043d\u044b\u043c \u2013 \u044d\u0442\u043e \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043f\u0435\u0440\u0435\u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u043c. \u0418\u043d\u0436\u0435\u043d\u0435\u0440\u043d\u043e\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u0435: \u0434\u0430\u0432\u0430\u0439\u0442\u0435 \u0434\u043e\u0431\u0430\u0432\u0438\u043c \u043a MSE-\u043b\u043e\u0441\u0441\u0443 \u0448\u0442\u0440\u0430\u0444 \u0437\u0430 \u0440\u0430\u0437\u043c\u0435\u0440 \u0432\u0435\u0441\u043e\u0432 \u2013 \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0437\u0430 \u0441\u0443\u043c\u043c\u0443 \u0438\u0445 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432, \u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"L_{2}\" alt=\"L_{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d84\/d843d71869425cb9eefb67d52e20c972.svg\" width=\"20\" height=\"15\" data-width=\"2.528\" data-height=\"1.885\" data-vertical-align=\"-0.339\" style=\"width: 2.528ex; height: 1.885ex; vertical-align: -0.339ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d84\/d843d71869425cb9eefb67d52e20c972.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d84\/d843d71869425cb9eefb67d52e20c972.svg 781w\" loading=\"lazy\" decode=\"async\"\/>-\u043d\u043e\u0440\u043c\u0443:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"MSE(\\theta) = \\frac{1}{2}\\left\\| Y - X\\theta \\right\\|_{2}^{2} + \\frac{\\lambda}{2}\\left\\| \\theta \\right\\|_{2}^{2},\\ \\ \\frac{d}{d\\theta} = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\widehat{\\theta} = \\left( X^{T}X + \\lambda I \\right)^{- 1}X^{T}Y\" alt=\"MSE(\\theta) = \\frac{1}{2}\\left\\| Y - X\\theta \\right\\|_{2}^{2} + \\frac{\\lambda}{2}\\left\\| \\theta \\right\\|_{2}^{2},\\ \\ \\frac{d}{d\\theta} = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\widehat{\\theta} = \\left( X^{T}X + \\lambda I \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07c\/07c01aadd8c05c594a31a503fec6dec4.svg\" width=\"583\" height=\"37\" data-width=\"72.93\" data-height=\"4.674\" data-vertical-align=\"-1.771\" style=\"width: 72.93ex; height: 4.674ex; vertical-align: -1.771ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07c\/07c01aadd8c05c594a31a503fec6dec4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/07\/07c\/07c01aadd8c05c594a31a503fec6dec4.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u043e\u0431\u0440\u0430\u0449\u0430\u0435\u043c\u0430\u044f \u043c\u0430\u0442\u0440\u0438\u0446\u0430 \u043e\u0442\u0434\u0435\u043b\u0438\u043b\u0430\u0441\u044c \u043e\u0442 \u0432\u044b\u0440\u043e\u0436\u0434\u0435\u043d\u043d\u043e\u0439, \u0441\u0434\u0435\u043b\u0430\u0432 \u043c\u043e\u0434\u0435\u043b\u044c \u0441\u0442\u0430\u0431\u0438\u043b\u044c\u043d\u0435\u0435.<\/p>\n<p>2) \u041f\u043e\u0434\u0445\u043e\u0434 \u0447\u0435\u0440\u0435\u0437 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f (\u041e\u041c\u041f):<\/p>\n<p>\u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c, \u0447\u0442\u043e \u0434\u0430\u043d\u043d\u044b\u0435 \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u043a\u0430\u043a <img decoding=\"async\" class=\"formula inline\" source=\"y_{i}\\ \\sim\\ N\\left( \\theta^{T}x_{i},\\sigma^{2} \\right)\" alt=\"y_{i}\\ \\sim\\ N\\left( \\theta^{T}x_{i},\\sigma^{2} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/45f\/45fd00b9c58428beb5e72a0de8f7f51e.svg\" width=\"145\" height=\"22\" data-width=\"18.15\" data-height=\"2.808\" data-vertical-align=\"-0.838\" style=\"width: 18.15ex; height: 2.808ex; vertical-align: -0.838ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/45f\/45fd00b9c58428beb5e72a0de8f7f51e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/45\/45f\/45fd00b9c58428beb5e72a0de8f7f51e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0414\u043b\u044f \u0432\u044b\u0431\u043e\u0440\u043a\u0438 \u0438\u0437 n \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u044b\u0445 \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u0439:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\ln{p\\left( Y\\ |\\ X,\\theta \\right)} = \\ln{\\prod_{i = 1}^{n}{\\frac{1}{\\sqrt{2\\pi\\sigma^{2}}}\\exp\\left( - \\frac{\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2}}{2\\sigma^{2}} \\right)}} = const - \\frac{1}{2\\sigma^{2}}\\sum_{i = 1}^{n}\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2}\" alt=\"\\ln{p\\left( Y\\ |\\ X,\\theta \\right)} = \\ln{\\prod_{i = 1}^{n}{\\frac{1}{\\sqrt{2\\pi\\sigma^{2}}}\\exp\\left( - \\frac{\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2}}{2\\sigma^{2}} \\right)}} = const - \\frac{1}{2\\sigma^{2}}\\sum_{i = 1}^{n}\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/539\/539e02dcb204a87573bba8c2fa9783d7.svg\" width=\"664\" height=\"56\" data-width=\"83.057\" data-height=\"7.028\" data-vertical-align=\"-2.949\" style=\"width: 83.057ex; height: 7.028ex; vertical-align: -2.949ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/539\/539e02dcb204a87573bba8c2fa9783d7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/53\/539\/539e02dcb204a87573bba8c2fa9783d7.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0434\u043b\u044f <img decoding=\"async\" class=\"formula inline\" source=\"MSE(\\theta)\" alt=\"MSE(\\theta)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e3116680e8cef8739f29e51e9ae4dc.svg\" width=\"67\" height=\"18\" data-width=\"8.387\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.387ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e3116680e8cef8739f29e51e9ae4dc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e3116680e8cef8739f29e51e9ae4dc.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0442\u043e\u043b\u044c\u043a\u043e \u0441 \u043c\u0438\u043d\u0443\u0441\u043e\u043c (\u0438\u0449\u0435\u043c \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u043e\u0448\u0438\u0431\u043a\u0438 vs \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f). \u041f\u0440\u043e\u0434\u043e\u043b\u0436\u0438\u043c \u0440\u0435\u0448\u0435\u043d\u0438\u0435:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\sum_{i = 1}^{n}\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2} = Y^{T}Y - 2\\theta^{T}X^{T}Y + \\theta^{T}X^{T}X\\theta\" alt=\"\\sum_{i = 1}^{n}\\left( y_{i} - \\theta^{T}x_{i} \\right)^{2} = Y^{T}Y - 2\\theta^{T}X^{T}Y + \\theta^{T}X^{T}X\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8d\/8dc\/8dc9ad5f27c479f938d83e05f9426699.svg\" width=\"372\" height=\"51\" data-width=\"46.487\" data-height=\"6.354\" data-vertical-align=\"-2.611\" style=\"width: 46.487ex; height: 6.354ex; vertical-align: -2.611ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8d\/8dc\/8dc9ad5f27c479f938d83e05f9426699.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8d\/8dc\/8dc9ad5f27c479f938d83e05f9426699.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"\\nabla_{\\theta}\\ln{p\\left( Y\\ |\\ X,\\theta \\right)} = - \\frac{1}{\\sigma^{2}}\\left( X^{T}X\\theta - X^{T}Y \\right) = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" alt=\"\\nabla_{\\theta}\\ln{p\\left( Y\\ |\\ X,\\theta \\right)} = - \\frac{1}{\\sigma^{2}}\\left( X^{T}X\\theta - X^{T}Y \\right) = 0\\ \\\u00a0 \\Rightarrow \\ \\ \\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/20f\/20f8d25d369f303ff2f8f5cdfcc0f92f.svg\" width=\"554\" height=\"38\" data-width=\"69.222\" data-height=\"4.69\" data-vertical-align=\"-1.779\" style=\"width: 69.222ex; height: 4.69ex; vertical-align: -1.779ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/20f\/20f8d25d369f303ff2f8f5cdfcc0f92f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/20f\/20f8d25d369f303ff2f8f5cdfcc0f92f.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u043d\u0430\u0448\u043b\u0438 \u0432 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u043e\u0446\u0435\u043d\u043a\u0443 \u043f\u043e \u041c\u041d\u041a \u2013 \u0441 \u0442\u043e\u0439 \u0436\u0435 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u043e\u0439 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0439 \u0432\u044b\u0440\u043e\u0436\u0434\u0435\u043d\u043d\u043e\u0441\u0442\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"X^{T}X\" alt=\"X^{T}X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d5\/5d5eaf3c0a410adace4935eb816a70a3.svg\" width=\"42\" height=\"16\" data-width=\"5.285\" data-height=\"2.017\" style=\"width: 5.285ex; height: 2.017ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d5\/5d5eaf3c0a410adace4935eb816a70a3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5d\/5d5\/5d5eaf3c0a410adace4935eb816a70a3.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u041f\u043e\u043a\u0430 \u0447\u0442\u043e \u043c\u044b \u043d\u0435 \u0443\u0447\u043b\u0438 \u0432\u0441\u044e \u0441\u0438\u043b\u0443 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u0411\u0430\u0439\u0435\u0441\u0430. \u0427\u0442\u043e \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0441\u043a\u0430\u0437\u0430\u0442\u044c \u043e \u0432\u0435\u0441\u0430\u0445 \u043c\u043e\u0434\u0435\u043b\u0438 \u0430\u043f\u0440\u0438\u043e\u0440\u043d\u043e, \u043f\u043e\u043a\u0430 \u043c\u044b \u0438\u0445 \u0435\u0449\u0435 \u043d\u0435 \u043e\u0446\u0435\u043d\u0438\u043b\u0438? \u0412 \u0446\u0435\u043b\u043e\u043c, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043e\u0442\u0442\u043e\u043b\u043a\u043d\u0443\u0442\u044c\u0441\u044f \u043e\u0442 \u043b\u044e\u0431\u043e\u0433\u043e \u0430\u043f\u0440\u0438\u043e\u0440\u0430, \u0438 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e\u0435 \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u0439 \u0435\u0433\u043e \u0432 \u043a\u043e\u043d\u0446\u0435 \u043a\u043e\u043d\u0446\u043e\u0432 \u0438\u0441\u043f\u0440\u0430\u0432\u044f\u0442. \u041d\u043e \u0440\u0430\u0437 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0431\u0438\u0440\u0430\u0442\u044c, \u0442\u043e \u0434\u0430\u0432\u0430\u0439\u0442\u0435 \u0432\u044b\u0431\u0435\u0440\u0435\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0441\u0438\u043b\u044c\u043d\u043e \u0443\u043f\u0440\u043e\u0441\u0442\u0438\u0442 \u0440\u0430\u0441\u0447\u0435\u0442\u044b: \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u044b\u0439 \u0430\u043f\u0440\u0438\u043e\u0440. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0435 \u043a \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u043c\u0443 \u2013 \u0442\u043e\u0436\u0435 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435. \u0414\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e: \u043c\u044b \u0432 \u0441\u0430\u043c\u043e\u043c \u043d\u0430\u0447\u0430\u043b\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u043b\u0438, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\psi(x) = \\frac{1}{2}x^{2}\" alt=\"\\psi(x) = \\frac{1}{2}x^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f7\/4f7d777ef50502accf02ee552365161d.svg\" width=\"96\" height=\"37\" data-width=\"11.953\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 11.953ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f7\/4f7d777ef50502accf02ee552365161d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f7\/4f7d777ef50502accf02ee552365161d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043d\u0435\u043f\u043e\u0434\u0432\u0438\u0436\u043d\u0430\u044f \u0442\u043e\u0447\u043a\u0430 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430, \u0430 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0438\u043c\u0435\u0435\u0442 \u0440\u043e\u0432\u043d\u043e \u0442\u0430\u043a\u043e\u0439 \u0436\u0435 \u0432\u0438\u0434. \u041f\u0443\u0441\u0442\u044c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\theta\\ \\sim\\ N\\left( \\theta_{0},\\Sigma_{0} \\right),\\ \\ p(\\theta) = \\frac{1}{(2\\pi)^{d\/2}\\left| \\Sigma_{0} \\right|^{1\/2}\\ }\\exp\\left( - \\frac{1}{2}\\left( \\theta - \\theta_{0} \\right)^{T}\\Sigma_{0}^{- 1}\\left( \\theta - \\theta_{0} \\right) \\right)\" alt=\"\\theta\\ \\sim\\ N\\left( \\theta_{0},\\Sigma_{0} \\right),\\ \\ p(\\theta) = \\frac{1}{(2\\pi)^{d\/2}\\left| \\Sigma_{0} \\right|^{1\/2}\\ }\\exp\\left( - \\frac{1}{2}\\left( \\theta - \\theta_{0} \\right)^{T}\\Sigma_{0}^{- 1}\\left( \\theta - \\theta_{0} \\right) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/51\/51e\/51eb9db8a1809216ed99e1a04595ea40.svg\" width=\"567\" height=\"48\" data-width=\"70.853\" data-height=\"6.033\" data-vertical-align=\"-2.451\" style=\"width: 70.853ex; height: 6.033ex; vertical-align: -2.451ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/51\/51e\/51eb9db8a1809216ed99e1a04595ea40.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/51\/51e\/51eb9db8a1809216ed99e1a04595ea40.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0411\u0430\u0439\u0435\u0441\u043e\u0432\u0441\u043a\u0438\u0439 \u0430\u043f\u0434\u0435\u0439\u0442:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p\\left( \\theta\\ |\\ X,Y \\right) = \\frac{p\\left( Y\\ |\\ X,\\theta \\right) \\cdot p(\\theta)}{p\\left( Y\\ |\\ X \\right)}\\\u00a0 \\propto \\ \\exp\\left( - \\frac{1}{2\\sigma^{2}}(Y - X\\theta)^{T}(Y - X\\theta) - \\frac{1}{2}\\left( \\theta - \\theta_{0} \\right)^{T}\\Sigma_{0}^{- 1}\\left( \\theta - \\theta_{0} \\right) \\right)\" alt=\"p\\left( \\theta\\ |\\ X,Y \\right) = \\frac{p\\left( Y\\ |\\ X,\\theta \\right) \\cdot p(\\theta)}{p\\left( Y\\ |\\ X \\right)}\\\u00a0 \\propto \\ \\exp\\left( - \\frac{1}{2\\sigma^{2}}(Y - X\\theta)^{T}(Y - X\\theta) - \\frac{1}{2}\\left( \\theta - \\theta_{0} \\right)^{T}\\Sigma_{0}^{- 1}\\left( \\theta - \\theta_{0} \\right) \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f3\/4f3a39e80ed5203dba3e5f134ba77051.svg\" width=\"756\" height=\"44\" data-width=\"94.492\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 94.492ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f3\/4f3a39e80ed5203dba3e5f134ba77051.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f3\/4f3a39e80ed5203dba3e5f134ba77051.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0410\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440\u043d\u044b\u0435 \u043a\u043e\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u044f \u0438 \u0441\u0440\u0435\u0434\u043d\u0435\u0435:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\Sigma_{n} = \\left( \\Sigma_{0}^{- 1} + \\frac{1}{\\sigma^{2}}X^{T}X \\right)^{- 1},\\ \\ \\theta_{n} = \\Sigma_{n}\\left( \\Sigma_{0}^{- 1}\\theta_{0} + \\frac{1}{\\sigma^{2}}X^{T}Y \\right)\" alt=\"\\Sigma_{n} = \\left( \\Sigma_{0}^{- 1} + \\frac{1}{\\sigma^{2}}X^{T}X \\right)^{- 1},\\ \\ \\theta_{n} = \\Sigma_{n}\\left( \\Sigma_{0}^{- 1}\\theta_{0} + \\frac{1}{\\sigma^{2}}X^{T}Y \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1f\/1fc\/1fc6183c90c167967317ffe6902191d4.svg\" width=\"460\" height=\"47\" data-width=\"57.443\" data-height=\"5.876\" data-vertical-align=\"-2.372\" style=\"width: 57.443ex; height: 5.876ex; vertical-align: -2.372ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1f\/1fc\/1fc6183c90c167967317ffe6902191d4.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1f\/1fc\/1fc6183c90c167967317ffe6902191d4.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0438\u043d\u0430\u0434\u043b\u0435\u0436\u0438\u0442 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u043c\u0443 \u043a\u043b\u0430\u0441\u0441\u0443, \u0438 \u0435\u0433\u043e \u043d\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u043e\u0447\u043d\u0430\u044f \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 \u0440\u0430\u0432\u043d\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"f(\\eta) = \\frac{1}{2}\\eta^{T}\\Sigma\\eta\" alt=\"f(\\eta) = \\frac{1}{2}\\eta^{T}\\Sigma\\eta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/06\/067\/067a56604ebd44bbdd62b2a24825aa19.svg\" width=\"116\" height=\"37\" data-width=\"14.469\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 14.469ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/06\/067\/067a56604ebd44bbdd62b2a24825aa19.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/06\/067\/067a56604ebd44bbdd62b2a24825aa19.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0433\u0434\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\eta = \\Sigma^{- 1}\\theta\" alt=\"\\eta = \\Sigma^{- 1}\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/baf\/baf2e2168f5a818ce0b3b0ddbb7e6a4e.svg\" width=\"73\" height=\"21\" data-width=\"9.068\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 9.068ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/baf\/baf2e2168f5a818ce0b3b0ddbb7e6a4e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/baf\/baf2e2168f5a818ce0b3b0ddbb7e6a4e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440. \u041f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430: <img decoding=\"async\" class=\"formula inline\" source=\"f^{*}(\\theta) = \\frac{1}{2}\\theta^{T}\\Sigma^{- 1}\\theta\" alt=\"f^{*}(\\theta) = \\frac{1}{2}\\theta^{T}\\Sigma^{- 1}\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/706\/70619f1778b5de26fd28f90e3b893fd1.svg\" width=\"141\" height=\"37\" data-width=\"17.619\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 17.619ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/706\/70619f1778b5de26fd28f90e3b893fd1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/706\/70619f1778b5de26fd28f90e3b893fd1.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e\u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c, \u043f\u0440\u0438 \u0431\u0430\u0439\u0435\u0441\u043e\u0432\u0441\u043a\u043e\u043c \u0430\u043f\u0434\u0435\u0439\u0442\u0435, \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440\u043d\u0430\u044f \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c (\u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u043a\u043e\u0432\u0430\u0440\u0438\u0430\u0446\u0438\u044f) \u0441\u0443\u043c\u043c\u0438\u0440\u0443\u0435\u0442\u0441\u044f: <img decoding=\"async\" class=\"formula inline\" source=\"\\Sigma_{n}^{- 1} = \\Sigma_{0}^{- 1} + \\frac{1}{\\sigma^{2}}X^{T}X\" alt=\"\\Sigma_{n}^{- 1} = \\Sigma_{0}^{- 1} + \\frac{1}{\\sigma^{2}}X^{T}X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e9\/e9e\/e9ec342628ec68862aa7d9044871d0b7.svg\" width=\"177\" height=\"38\" data-width=\"22.074\" data-height=\"4.69\" data-vertical-align=\"-1.779\" style=\"width: 22.074ex; height: 4.69ex; vertical-align: -1.779ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e9\/e9e\/e9ec342628ec68862aa7d9044871d0b7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e9\/e9e\/e9ec342628ec68862aa7d9044871d0b7.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0442.\u0435. \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u0438 \u0424\u0438\u0448\u0435\u0440\u0430 \u0432 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435.<\/p>\n<p>\u041f\u043e\u0441\u0447\u0438\u0442\u0430\u0432 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta_{n}\" alt=\"\\theta_{n}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f54\/f544827628869ea2e44757fd4040da94.svg\" width=\"18\" height=\"18\" data-width=\"2.209\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 2.209ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f54\/f544827628869ea2e44757fd4040da94.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f54\/f544827628869ea2e44757fd4040da94.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u0442.\u0435. \u043f\u0440\u043e\u0432\u0435\u0434\u044f \u0430\u043f\u0434\u0435\u0439\u0442 \u043f\u043e \u0432\u0441\u0435\u043c\u0443 \u0434\u0430\u0442\u0430\u0441\u0435\u0442\u0443), \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u0443\u0442\u043e\u0447\u043d\u0435\u043d\u043d\u0443\u044e \u043e\u0446\u0435\u043d\u043a\u0443 \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta}\" alt=\"\\widehat{\\theta}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg\" width=\"12\" height=\"24\" data-width=\"1.061\" data-height=\"2.993\" data-vertical-align=\"-0.931\" style=\"width: 1.061ex; height: 2.993ex; vertical-align: -0.931ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\widehat{\\theta} = \\left( \\Sigma_{0}^{- 1} + \\frac{1}{\\sigma^{2}}X^{T}X \\right)^{- 1}\\left( \\Sigma_{0}^{- 1}\\theta_{0} + \\frac{1}{\\sigma^{2}}X^{T}Y \\right)\" alt=\"\\widehat{\\theta} = \\left( \\Sigma_{0}^{- 1} + \\frac{1}{\\sigma^{2}}X^{T}X \\right)^{- 1}\\left( \\Sigma_{0}^{- 1}\\theta_{0} + \\frac{1}{\\sigma^{2}}X^{T}Y \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/20e\/20e98f36dab4a99ce4f160ff5350f6f0.svg\" width=\"365\" height=\"47\" data-width=\"45.579\" data-height=\"5.876\" data-vertical-align=\"-2.372\" style=\"width: 45.579ex; height: 5.876ex; vertical-align: -2.372ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/20e\/20e98f36dab4a99ce4f160ff5350f6f0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/20\/20e\/20e98f36dab4a99ce4f160ff5350f6f0.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u0432\u044b\u0431\u0438\u0440\u0430\u043b\u0438 \u0430\u043f\u0440\u0438\u043e\u0440 \u0441\u0430\u043c\u0438, \u043f\u0443\u0441\u0442\u044c \u043e\u043d \u0431\u0443\u0434\u0435\u0442 \u0443\u0434\u043e\u0431\u043d\u044b\u043c, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 <img decoding=\"async\" class=\"formula inline\" source=\"\\Sigma_{0} = \\sigma_{0}^{2}I\" alt=\"\\Sigma_{0} = \\sigma_{0}^{2}I\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/81\/810\/810e68f252317e9c36474c6e6ec15563.svg\" width=\"72\" height=\"21\" data-width=\"9.058\" data-height=\"2.594\" data-vertical-align=\"-0.731\" style=\"width: 9.058ex; height: 2.594ex; vertical-align: -0.731ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/81\/810\/810e68f252317e9c36474c6e6ec15563.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/81\/810\/810e68f252317e9c36474c6e6ec15563.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta_{0} = 0\" alt=\"\\theta_{0} = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0e\/0ed\/0edc6a232397ea2ecaa8f35d471eebbc.svg\" width=\"50\" height=\"18\" data-width=\"6.197\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.197ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0e\/0ed\/0edc6a232397ea2ecaa8f35d471eebbc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0e\/0ed\/0edc6a232397ea2ecaa8f35d471eebbc.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0422\u043e\u0433\u0434\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\widehat{\\theta} = \\left( X^{T}X + \\frac{\\sigma^{2}}{\\sigma_{0}^{2}}I \\right)^{- 1}X^{T}Y\" alt=\"\\widehat{\\theta} = \\left( X^{T}X + \\frac{\\sigma^{2}}{\\sigma_{0}^{2}}I \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0c\/0cf\/0cf0d430e16c5dc7807211ae0657c616.svg\" width=\"220\" height=\"58\" data-width=\"27.444\" data-height=\"7.233\" data-vertical-align=\"-3.051\" style=\"width: 27.444ex; height: 7.233ex; vertical-align: -3.051ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0c\/0cf\/0cf0d430e16c5dc7807211ae0657c616.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0c\/0cf\/0cf0d430e16c5dc7807211ae0657c616.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u0443\u0436\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u043b\u0438 \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435, \u043a\u043e\u0433\u0434\u0430 \u0440\u0435\u0433\u0443\u043b\u044f\u0440\u0438\u0437\u043e\u0432\u044b\u0432\u0430\u043b\u0438 \u043b\u043e\u0441\u0441 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u043e\u0439 \u043d\u043e\u0440\u043c\u043e\u0439 \u0432\u0435\u0441\u043e\u0432. \u041f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f, \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442 \u0440\u0435\u0433\u0443\u043b\u044f\u0440\u0438\u0437\u0430\u0446\u0438\u0438 \u0442\u0430\u043c \u2013 \u044d\u0442\u043e \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u0430\u043f\u0440\u0438\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043e \u0441\u043e\u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0438 \u0434\u0438\u0441\u043f\u0435\u0440\u0441\u0438\u0438 \u0448\u0443\u043c\u043e\u0432 \u0432 \u043c\u0435\u0442\u043a\u0430\u0445 \u0438 \u0432\u0435\u0441\u0430\u0445:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\widehat{\\theta} = \\left( X^{T}X + \\lambda I \\right)^{- 1}X^{T}Y,\\ \\ \\lambda = \\frac{\\sigma^{2}}{\\sigma_{0}^{2}}\" alt=\"\\widehat{\\theta} = \\left( X^{T}X + \\lambda I \\right)^{- 1}X^{T}Y,\\ \\ \\lambda = \\frac{\\sigma^{2}}{\\sigma_{0}^{2}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e68\/e68b3512f6a45ab2aa7fd67bae53fa98.svg\" width=\"270\" height=\"47\" data-width=\"33.725\" data-height=\"5.899\" data-vertical-align=\"-2.384\" style=\"width: 33.725ex; height: 5.899ex; vertical-align: -2.384ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e68\/e68b3512f6a45ab2aa7fd67bae53fa98.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e68\/e68b3512f6a45ab2aa7fd67bae53fa98.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u043e\u0442\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \u0441\u0432\u044f\u0437\u044c MSE-\u043b\u043e\u0441\u0441\u0430 \u0438 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f \u0447\u0443\u0442\u044c \u0433\u043b\u0443\u0431\u0436\u0435. \u041f\u0443\u0441\u0442\u044c \u0432\u0435\u043a\u0442\u043e\u0440 \u043e\u0448\u0438\u0431\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\varepsilon = Y - X\\theta\" alt=\"\\varepsilon = Y - X\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e8\/e8c\/e8cc0510e3379adf855fa4115eb7df69.svg\" width=\"92\" height=\"18\" data-width=\"11.552\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 11.552ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e8\/e8c\/e8cc0510e3379adf855fa4115eb7df69.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e8\/e8c\/e8cc0510e3379adf855fa4115eb7df69.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442\u043e\u0433\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"MSE(\\theta) = \\frac{1}{2}\\left\\| \\varepsilon \\right\\|_{2}^{2}\" alt=\"MSE(\\theta) = \\frac{1}{2}\\left\\| \\varepsilon \\right\\|_{2}^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f7\/f75\/f75908d77a9f0bde152d7474132a9e26.svg\" width=\"153\" height=\"37\" data-width=\"19.093\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 19.093ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f7\/f75\/f75908d77a9f0bde152d7474132a9e26.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f7\/f75\/f75908d77a9f0bde152d7474132a9e26.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u0435\u0440\u0435\u0439\u0434\u0435\u043c \u043a \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0435 \u0447\u0435\u0440\u0435\u0437 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\varepsilon\" alt=\"\\varepsilon\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8b\/f8b1c5a729a09649c275fca88976d8dd.svg\" width=\"12\" height=\"18\" data-width=\"1.054\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.054ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8b\/f8b1c5a729a09649c275fca88976d8dd.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8b\/f8b1c5a729a09649c275fca88976d8dd.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0421\u043d\u043e\u0432\u0430 \u0432\u0441\u043f\u043e\u043c\u043d\u0438\u043c, \u0447\u0442\u043e <img decoding=\"async\" class=\"formula inline\" source=\"\\psi(x) = \\frac{1}{2}x^{2}\" alt=\"\\psi(x) = \\frac{1}{2}x^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f7\/4f7d777ef50502accf02ee552365161d.svg\" width=\"96\" height=\"37\" data-width=\"11.953\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 11.953ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f7\/4f7d777ef50502accf02ee552365161d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f7\/4f7d777ef50502accf02ee552365161d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043d\u0435\u043f\u043e\u0434\u0432\u0438\u0436\u043d\u0430\u044f \u0442\u043e\u0447\u043a\u0430 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"MSE^{*}(\\lambda) = \\frac{1}{2}\\left\\| \\lambda \\right\\|_{2}^{2},\" alt=\"MSE^{*}(\\lambda) = \\frac{1}{2}\\left\\| \\lambda \\right\\|_{2}^{2},\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea7f5d516b20be46599ea11e4ba95e8e.svg\" width=\"161\" height=\"37\" data-width=\"20.09\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 20.09ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea7f5d516b20be46599ea11e4ba95e8e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea7\/ea7f5d516b20be46599ea11e4ba95e8e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0433\u0434\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda\" alt=\"\\lambda\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg\" width=\"12\" height=\"18\" data-width=\"1.319\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.319ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c6\/c6a\/c6a6eb61fd9c6c913da73b3642ca147d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u0430\u044f. \u0420\u0435\u0448\u0435\u043d\u0438\u0435 \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438 \u0438\u043c\u0435\u0435\u0442 \u0442\u043e\u0442 \u0436\u0435 \u0432\u0438\u0434. \u0411\u043e\u043b\u0435\u0435 \u0442\u043e\u0433\u043e, \u0442\u0435\u043f\u0435\u0440\u044c \u0440\u0435\u0448\u0435\u043d\u0438\u0435 \u0431\u0435\u0437 \u0440\u0435\u0433\u0443\u043b\u044f\u0440\u0438\u0437\u0430\u0446\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" alt=\"\\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b4\/2b4ad7e3547e7de66c1aac7744b31db2.svg\" width=\"150\" height=\"26\" data-width=\"18.752\" data-height=\"3.256\" data-vertical-align=\"-1.062\" style=\"width: 18.752ex; height: 3.256ex; vertical-align: -1.062ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b4\/2b4ad7e3547e7de66c1aac7744b31db2.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b4\/2b4ad7e3547e7de66c1aac7744b31db2.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u044f\u0432\u043d\u043e \u0432\u044b\u0445\u043e\u0434\u0438\u0442 \u043a\u0430\u043a \u0447\u0430\u0441\u0442\u043d\u044b\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u043f\u0440\u0438 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"X^{T}\\lambda = 0\" alt=\"X^{T}\\lambda = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b0\/7b09ddf9037d90dfec761c72fee6ce14.svg\" width=\"71\" height=\"21\" data-width=\"8.825\" data-height=\"2.583\" data-vertical-align=\"-0.726\" style=\"width: 8.825ex; height: 2.583ex; vertical-align: -0.726ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b0\/7b09ddf9037d90dfec761c72fee6ce14.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b0\/7b09ddf9037d90dfec761c72fee6ce14.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>3) \u0418\u0442\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u044b\u0439 \u043f\u043e\u0434\u0445\u043e\u0434:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\theta_{t + 1} = \\theta_{t} - \\eta\\nabla MSE\\left( \\theta_{t} \\right),\\ \\ \u0433\u0434\u0435\\ \\nabla MSE\\left( \\theta_{t} \\right) = X^{T}X\\theta_{t} - X^{T}Y\" alt=\"\\theta_{t + 1} = \\theta_{t} - \\eta\\nabla MSE\\left( \\theta_{t} \\right),\\ \\ \u0433\u0434\u0435\\ \\nabla MSE\\left( \\theta_{t} \\right) = X^{T}X\\theta_{t} - X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c0e\/c0e1f679b619027a481026e2ec0b8d61.svg\" width=\"481\" height=\"21\" data-width=\"60.186\" data-height=\"2.583\" data-vertical-align=\"-0.726\" style=\"width: 60.186ex; height: 2.583ex; vertical-align: -0.726ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c0e\/c0e1f679b619027a481026e2ec0b8d61.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c0e\/c0e1f679b619027a481026e2ec0b8d61.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u043e\u0441\u0442\u0430\u043d\u0430\u0432\u043b\u0438\u0432\u0430\u0435\u0442\u0441\u044f, \u043a\u043e\u0433\u0434\u0430 \u0432\u044b\u0445\u043e\u0434\u0438\u0442 \u043d\u0430 \u043f\u043b\u0430\u0442\u043e, \u0442.\u0435. <img decoding=\"async\" class=\"formula inline\" source=\"0 = \\nabla MSE(\\theta) = X^{T}X\\theta - X^{T}Y\" alt=\"0 = \\nabla MSE(\\theta) = X^{T}X\\theta - X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbabf63fb1d26a6014649ae9882050db.svg\" width=\"253\" height=\"21\" data-width=\"31.632\" data-height=\"2.583\" data-vertical-align=\"-0.726\" style=\"width: 31.632ex; height: 2.583ex; vertical-align: -0.726ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbabf63fb1d26a6014649ae9882050db.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbabf63fb1d26a6014649ae9882050db.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043e\u0442\u043a\u0443\u0434\u0430 \u0441\u043d\u043e\u0432\u0430 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u0440\u0435\u0448\u0435\u043d\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" alt=\"\\widehat{\\theta} = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b4\/2b4ad7e3547e7de66c1aac7744b31db2.svg\" width=\"150\" height=\"26\" data-width=\"18.752\" data-height=\"3.256\" data-vertical-align=\"-1.062\" style=\"width: 18.752ex; height: 3.256ex; vertical-align: -1.062ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b4\/2b4ad7e3547e7de66c1aac7744b31db2.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b4\/2b4ad7e3547e7de66c1aac7744b31db2.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0412\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u043d\u043e\u0432\u0430\u044f \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430 &#8212; \u0442\u0435\u043f\u0435\u0440\u044c \u0441 \u0443\u0447\u0435\u0442\u043e\u043c \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 \u0432\u0435\u0441\u043e\u0432: \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0438 \u043c\u043e\u0433\u0443\u0442 \u0438\u043c\u0435\u0442\u044c \u0440\u0430\u0437\u043d\u044b\u0439 \u043c\u0430\u0441\u0448\u0442\u0430\u0431, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u043e\u0447\u0435\u043d\u044c \u0434\u043e\u043b\u0433\u043e \u0441\u0445\u043e\u0434\u0438\u0442\u0441\u044f. \u041a\u043e\u0433\u0434\u0430-\u0442\u043e \u0434\u043b\u044f \u0443\u0447\u0435\u0442\u0430 \u043a\u043e\u0440\u0440\u0435\u043a\u0442\u043d\u043e\u0439 \u043c\u0435\u0442\u0440\u0438\u043a\u0438 \u043c\u044b \u043f\u0440\u0438\u0434\u0443\u043c\u0430\u043b\u0438 \u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u2013 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c \u0435\u0433\u043e \u0437\u0434\u0435\u0441\u044c. \u041c\u044b \u0443\u0436\u0435 \u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e MSE \u2013 \u044d\u0442\u043e \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"I(\\theta)\\mathbb{= - E}\\nabla^{2}l(\\theta) = \\nabla\\frac{1}{\\sigma^{2}}\\left( X^{T}X\\theta - X^{T}Y \\right) = \\frac{1}{\\sigma^{2}}X^{T}X\\ \\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ \\ \\widetilde{\\nabla}\\\u00a0 = \\left( I(\\theta) \\right)^{- 1}\\nabla\\\u00a0 = \\sigma^{2}\\left( X^{T}X \\right)^{- 1}\" alt=\"I(\\theta)\\mathbb{= - E}\\nabla^{2}l(\\theta) = \\nabla\\frac{1}{\\sigma^{2}}\\left( X^{T}X\\theta - X^{T}Y \\right) = \\frac{1}{\\sigma^{2}}X^{T}X\\ \\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ \\ \\widetilde{\\nabla}\\\u00a0 = \\left( I(\\theta) \\right)^{- 1}\\nabla\\\u00a0 = \\sigma^{2}\\left( X^{T}X \\right)^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d46\/d464c4166ff75d0a0d227cb9f9b7285d.svg\" width=\"731\" height=\"38\" data-width=\"91.336\" data-height=\"4.69\" data-vertical-align=\"-1.779\" style=\"width: 91.336ex; height: 4.69ex; vertical-align: -1.779ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d46\/d464c4166ff75d0a0d227cb9f9b7285d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d4\/d46\/d464c4166ff75d0a0d227cb9f9b7285d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u043e\u0437\u044c\u043c\u0435\u043c \u0443\u0434\u043e\u0431\u043d\u044b\u0439 \u0440\u0430\u0437\u043c\u0435\u0440 \u0448\u0430\u0433\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\eta = \\frac{1}{\\sigma^{2}}\" alt=\"\\eta = \\frac{1}{\\sigma^{2}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d3\/6d3b0b02a7d1a5f4d1912cbc620f6126.svg\" width=\"59\" height=\"38\" data-width=\"7.417\" data-height=\"4.69\" data-vertical-align=\"-1.779\" style=\"width: 7.417ex; height: 4.69ex; vertical-align: -1.779ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d3\/6d3b0b02a7d1a5f4d1912cbc620f6126.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d3\/6d3b0b02a7d1a5f4d1912cbc620f6126.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\theta_{t + 1} = \\theta_{t} - \\eta\\sigma^{2}\\left( X^{T}X \\right)^{- 1}\\left( X^{T}X\\theta_{t} - X^{T}Y \\right) = \\theta_{t} - \\theta_{t} + \\left( X^{T}X \\right)^{- 1}X^{T}Y = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" alt=\"\\theta_{t + 1} = \\theta_{t} - \\eta\\sigma^{2}\\left( X^{T}X \\right)^{- 1}\\left( X^{T}X\\theta_{t} - X^{T}Y \\right) = \\theta_{t} - \\theta_{t} + \\left( X^{T}X \\right)^{- 1}X^{T}Y = \\left( X^{T}X \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/75\/756\/7565f657cfd62717d8780b02125df12f.svg\" width=\"692\" height=\"26\" data-width=\"86.45\" data-height=\"3.256\" data-vertical-align=\"-1.062\" style=\"width: 86.45ex; height: 3.256ex; vertical-align: -1.062ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/75\/756\/7565f657cfd62717d8780b02125df12f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/75\/756\/7565f657cfd62717d8780b02125df12f.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0422\u043e \u0435\u0441\u0442\u044c \u043f\u0440\u043e\u0441\u0442\u043e \u0443\u0447\u0442\u044f \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430, \u043c\u044b \u0432\u0441\u0435\u0433\u043e \u0437\u0430 1 \u0448\u0430\u0433 \u043f\u0435\u0440\u0435\u0448\u043b\u0438 \u0432 \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u0435. \u041f\u043e\u0447\u0435\u043c\u0443 \u0442\u0430\u043a \u043f\u0440\u043e\u0438\u0437\u043e\u0448\u043b\u043e? \u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c \u043c\u0435\u0442\u043e\u0434 \u041d\u044c\u044e\u0442\u043e\u043d\u0430: \u0431\u0435\u0440\u0435\u043c \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0438\u043c \u0432 \u0435\u0433\u043e \u043c\u0438\u043d\u0438\u043c\u0443\u043c. \u041d\u043e MSE \u0441\u0430\u043c\u0430 \u043f\u043e \u0441\u0435\u0431\u0435 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u0430, \u0442.\u0435. \u0440\u0430\u0432\u043d\u0430 \u0441\u0432\u043e\u0435\u043c\u0443 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u043e\u043c\u0443 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044e, \u0438 \u043f\u0435\u0440\u0435\u0439\u0434\u044f \u0432 \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044f, \u043c\u044b \u043f\u0435\u0440\u0435\u0448\u043b\u0438 \u0432 \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0432\u0441\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043d\u044b\u0439 \u0441\u043f\u0443\u0441\u043a \u043f\u043e MSE \u0441 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u043c \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043e\u043c \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u0435\u043d \u043c\u0435\u0442\u043e\u0434\u0443 \u041d\u044c\u044e\u0442\u043e\u043d\u0430.<\/p>\n<p>\u041e\u0441\u0442\u0430\u0435\u0442\u0441\u044f \u0434\u0432\u0430 \u0444\u0438\u043d\u0430\u043b\u044c\u043d\u044b\u0445 \u0432\u043e\u043f\u0440\u043e\u0441\u0430: \u043f\u043e\u0447\u0435\u043c\u0443 \u043c\u044b \u0432\u044b\u0431\u0440\u0430\u043b\u0438 \u0438\u043c\u0435\u043d\u043d\u043e <img decoding=\"async\" class=\"formula inline\" source=\"L_{2}\" alt=\"L_{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d84\/d843d71869425cb9eefb67d52e20c972.svg\" width=\"20\" height=\"15\" data-width=\"2.528\" data-height=\"1.885\" data-vertical-align=\"-0.339\" style=\"width: 2.528ex; height: 1.885ex; vertical-align: -0.339ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d84\/d843d71869425cb9eefb67d52e20c972.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d84\/d843d71869425cb9eefb67d52e20c972.svg 781w\" loading=\"lazy\" decode=\"async\"\/>-\u043d\u043e\u0440\u043c\u0443 \u0432\u0435\u0441\u043e\u0432 \u0438 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u044b\u0439 MSE-\u043b\u043e\u0441\u0441? \u041d\u0430 \u043f\u0435\u0440\u0432\u044b\u0439 \u043c\u044b \u0443\u0436\u0435 \u043e\u0442\u0432\u0435\u0442\u0438\u043b\u0438 \u2013 \u044d\u0442\u043e \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044e \u043e \u0442\u043e\u043c, \u0447\u0442\u043e \u0430\u043f\u0440\u0438\u043e\u0440\u043d\u043e \u0432\u0435\u0441\u0430 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u044b \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e. \u041c\u044b \u043c\u043e\u0433\u043b\u0438 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"L_{1}\" alt=\"L_{1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d7\/d74\/d748e209c28a70d71adff098dde47f2c.svg\" width=\"20\" height=\"15\" data-width=\"2.528\" data-height=\"1.885\" data-vertical-align=\"-0.339\" style=\"width: 2.528ex; height: 1.885ex; vertical-align: -0.339ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d7\/d74\/d748e209c28a70d71adff098dde47f2c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d7\/d74\/d748e209c28a70d71adff098dde47f2c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>-\u043d\u043e\u0440\u043c\u0443 \u0432\u0435\u0441\u043e\u0432 (\u0441\u0443\u043c\u043c\u0443 \u043c\u043e\u0434\u0443\u043b\u0435\u0439) \u2013 \u044d\u0442\u043e \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044e \u043e \u0442\u043e\u043c, \u0447\u0442\u043e \u0430\u043f\u0440\u0438\u043e\u0440\u043d\u043e \u0432\u0435\u0441\u0430 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u044b \u043f\u043e \u041b\u0430\u043f\u043b\u0430\u0441\u0443. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u043e\u0442\u0432\u0435\u0442 \u043d\u0430 \u0432\u0442\u043e\u0440\u043e\u0439 \u0432\u043e\u043f\u0440\u043e\u0441 \u2013 \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u044b\u0439: MSE-\u043b\u043e\u0441\u0441 \u043a\u043e\u0434\u0438\u0440\u0443\u0435\u0442 \u0430\u043f\u0440\u0438\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435, \u0447\u0442\u043e \u0434\u0430\u043d\u043d\u044b\u0435 \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u0441 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u044b\u043c \u0448\u0443\u043c\u043e\u043c, \u0430 MAE-\u043b\u043e\u0441\u0441 (\u0441\u0443\u043c\u043c\u0430 \u043c\u043e\u0434\u0443\u043b\u0435\u0439 \u043e\u0448\u0438\u0431\u043e\u043a) \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u0435\u043d \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044e \u043e \u043b\u0430\u043f\u043b\u0430\u0441\u0441\u043e\u0432\u0441\u043a\u043e\u043c \u0448\u0443\u043c\u0435. \u0424\u0443\u043d\u043a\u0446\u0438\u044f \u043c\u043e\u0434\u0443\u043b\u044f \u0445\u043e\u0442\u044c \u0438 \u043d\u0435 \u0433\u043b\u0430\u0434\u043a\u0430\u044f \u0432 \u043d\u0443\u043b\u0435, \u043d\u043e \u0432\u044b\u043f\u0443\u043a\u043b\u0430\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0432\u0435\u0437\u0434\u0435 \u0442\u043e\u0436\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u0438\u043c\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u2013 \u0441 \u043f\u0430\u0440\u043e\u0439 \u043e\u0433\u043e\u0432\u043e\u0440\u043e\u043a.<\/p>\n<p>\u041b\u043e\u0433\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u044f, \u043d\u0435\u0439\u0440\u043e\u043d, SVM<\/p>\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0437\u0430\u0434\u0430\u0447\u0443 \u043a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u0438: \u0434\u0430\u043d\u044b \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u044b\u0435 (\u0438\u043b\u0438 \u0431\u0438\u043d\u0430\u0440\u043d\u044b\u0435) \u043c\u0435\u0442\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"Y\" alt=\"Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg\" width=\"14\" height=\"12\" data-width=\"1.726\" data-height=\"1.545\" style=\"width: 1.726ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u0447\u0438\u0441\u043b\u043e\u0432\u044b\u0435 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"X\" alt=\"X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg\" width=\"15\" height=\"12\" data-width=\"1.928\" data-height=\"1.545\" style=\"width: 1.928ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u043c\u0435\u0442\u043a\u0438 \u043f\u043e\u0440\u043e\u0436\u0434\u0435\u043d\u044b \u0438\u0437 \u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f (\u0438\u043b\u0438 \u0411\u0435\u0440\u043d\u0443\u043b\u043b\u0438). \u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u043c \u043f\u043e\u0434\u043e\u0431\u0440\u0430\u0442\u044c \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0435 \u0432\u0435\u0441\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\widehat{\\theta}\" alt=\"\\widehat{\\theta}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg\" width=\"12\" height=\"24\" data-width=\"1.061\" data-height=\"2.993\" data-vertical-align=\"-0.931\" style=\"width: 1.061ex; height: 2.993ex; vertical-align: -0.931ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/2b\/2b5\/2b5c5c798d1539b9837e89a745afc42a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043c\u043e\u0434\u0435\u043b\u0438. \u0412 \u0431\u0438\u043d\u0430\u0440\u043d\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u044f \u0442\u0438\u043f\u0438\u0447\u043d\u0430 \u043c\u043e\u0434\u0435\u043b\u044c \u043b\u043e\u0433\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"y = \\sigma\\left( \\theta^{T}x \\right)\" alt=\"y = \\sigma\\left( \\theta^{T}x \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cae\/caeffed7b9e5b20f16109939fc70397f.svg\" width=\"92\" height=\"22\" data-width=\"11.536\" data-height=\"2.808\" data-vertical-align=\"-0.838\" style=\"width: 11.536ex; height: 2.808ex; vertical-align: -0.838ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cae\/caeffed7b9e5b20f16109939fc70397f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cae\/caeffed7b9e5b20f16109939fc70397f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e \u0441\u0443\u0442\u0438, \u043c\u044b \u0433\u043e\u0432\u043e\u0440\u0438\u043c, \u0447\u0442\u043e, \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439 \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u0438 \u043c\u044b \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0443\u0435\u043c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0448\u0430\u043d\u0441\u044b (\u043b\u043e\u0433\u0438\u0442 \u2013 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440 \u0411\u0435\u0440\u043d\u0443\u043b\u043b\u0438), \u0430 \u043f\u043e\u0442\u043e\u043c \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u043c \u0441\u0438\u0433\u043c\u043e\u0438\u0434\u0443 \u0438 \u043e\u0442\u043e\u0431\u0440\u0430\u0436\u0430\u0435\u043c \u0432 \u043a\u043e\u0440\u0440\u0435\u043a\u0442\u043d\u044b\u0435 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 \u043a\u043b\u0430\u0441\u0441\u043e\u0432. \u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u044f \u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e (\u0438\u043b\u0438 \u0411\u0435\u0440\u043d\u0443\u043b\u043b\u0438) \u2013 \u044d\u0442\u043e \u043a\u0440\u043e\u0441\u0441-\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f: <img decoding=\"async\" class=\"formula inline\" source=\"- \\sum_{cls}^{}{y_{cls}\\ln p_{cls}}\" alt=\"- \\sum_{cls}^{}{y_{cls}\\ln p_{cls}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30e\/30edb95861293c6adc5e05ba0e837003.svg\" width=\"119\" height=\"45\" data-width=\"14.884\" data-height=\"5.578\" data-vertical-align=\"-2.223\" style=\"width: 14.884ex; height: 5.578ex; vertical-align: -2.223ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30e\/30edb95861293c6adc5e05ba0e837003.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30e\/30edb95861293c6adc5e05ba0e837003.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0421\u0432\u044f\u0437\u043a\u0438 \u044d\u0442\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438 \u0441 \u0444\u0438\u0437\u0438\u043a\u043e\u0439 \u0438 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043c\u044b \u0434\u0435\u0442\u0430\u043b\u044c\u043d\u043e \u0438\u0437\u0443\u0447\u0438\u043b\u0438 \u0440\u0430\u0437\u0434\u0435\u043b\u0430 \u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0430.<\/p>\n<p>\u041a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u043d\u0435\u0439\u0440\u043e\u0441\u0435\u0442\u044c \u0441\u043e\u0441\u0442\u043e\u0438\u0442 \u0438\u0437 \u043d\u0430\u0431\u043e\u0440\u0430 \u0441\u043b\u043e\u0435\u0432, \u043a\u0430\u0436\u0434\u044b\u0439 \u0441\u043b\u043e\u0439 \u2013 \u0438\u0437 \u043d\u0435\u0439\u0440\u043e\u043d\u043e\u0432, \u0430 \u043d\u0435\u0439\u0440\u043e\u043d \u2013 \u044d\u0442\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432\u0438\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"y = \\sigma\\left( \\theta^{T}x \\right)\" alt=\"y = \\sigma\\left( \\theta^{T}x \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cae\/caeffed7b9e5b20f16109939fc70397f.svg\" width=\"92\" height=\"22\" data-width=\"11.536\" data-height=\"2.808\" data-vertical-align=\"-0.838\" style=\"width: 11.536ex; height: 2.808ex; vertical-align: -0.838ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cae\/caeffed7b9e5b20f16109939fc70397f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/cae\/caeffed7b9e5b20f16109939fc70397f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0433\u0434\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\sigma\" alt=\"\\sigma\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a2a\/a2ab7d71a0f07f388ff823293c147d21.svg\" width=\"12\" height=\"18\" data-width=\"1.292\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.292ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a2a\/a2ab7d71a0f07f388ff823293c147d21.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a2\/a2a\/a2ab7d71a0f07f388ff823293c147d21.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0430\u043a\u0442\u0438\u0432\u0430\u0446\u0438\u0438 (\u0442\u0438\u043f\u0438\u0447\u043d\u043e \u044d\u0442\u043e \u0441\u0438\u0433\u043c\u043e\u0438\u0434\u0430, \u0438 \u0442\u043e\u0433\u0434\u0430 \u043d\u0435\u0439\u0440\u043e\u043d \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0435\u0442 \u0441 \u043b\u043e\u0433\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u0435\u0439). \u041e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u043f\u043e \u0448\u0430\u0433\u0430\u043c:<\/p>\n<ol>\n<li>\n<p>forward-pass: \u043d\u0435\u0439\u0440\u043e\u043d \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442 \u043d\u0430 \u0432\u0445\u043e\u0434 \u0442\u043e\u0447\u043a\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"x\" alt=\"x\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg\" width=\"12\" height=\"12\" data-width=\"1.294\" data-height=\"1.025\" data-vertical-align=\"-0.025\" style=\"width: 1.294ex; height: 1.025ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041b\u0438\u043d\u0435\u0439\u043d\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u0438\u0442 \u0435\u0433\u043e \u0432 \u043b\u043e\u0433\u0438\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"z = \\theta^{T}x\" alt=\"z = \\theta^{T}x\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b8\/8b84966bc12b67f9cd355f0fef532665.svg\" width=\"62\" height=\"21\" data-width=\"7.738\" data-height=\"2.583\" data-vertical-align=\"-0.726\" style=\"width: 7.738ex; height: 2.583ex; vertical-align: -0.726ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b8\/8b84966bc12b67f9cd355f0fef532665.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8b\/8b8\/8b84966bc12b67f9cd355f0fef532665.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u043b\u0435\u043c\u0435\u043d\u0442 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0433\u043e \u043a \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044f\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430, \u0430\u043a\u0442\u0438\u0432\u0430\u0446\u0438\u044f (\u043a\u0430\u043a <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla\\psi(z)\" alt=\"\\nabla\\psi(z)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d6\/d60\/d60fa921c8810447c3de8f78715a786d.svg\" width=\"49\" height=\"18\" data-width=\"6.17\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.17ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d6\/d60\/d60fa921c8810447c3de8f78715a786d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d6\/d60\/d60fa921c8810447c3de8f78715a786d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>) \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u0438\u0442 \u043b\u043e\u0433\u0438\u0442 \u0432 \u043f\u0435\u0440\u0432\u0438\u0447\u043d\u043e\u0435 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439\/\u0441\u0440\u0435\u0434\u043d\u0438\u0445. \u0421\u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u043b\u043e\u0441\u0441 \u2013 \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432 \u0437\u0430\u0437\u043e\u0440<\/p>\n<\/li>\n<li>\n<p>backward-pass: \u0440\u0430\u0441\u0447\u0435\u0442 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043e\u0432 \u0432\u0435\u0441\u043e\u0432 \u043f\u043e \u0446\u0435\u043f\u043e\u0447\u043d\u043e\u043c\u0443 \u043f\u0440\u0430\u0432\u0438\u043b\u0443, \u043e\u0442 \u043b\u043e\u0441\u0441\u0430 \u043d\u0430\u0437\u0430\u0434 \u043f\u043e \u0432\u0441\u0435\u043c \u0441\u043b\u043e\u044f\u043c. \u0413\u0440\u0430\u0434\u0438\u0435\u043d\u0442 \u043b\u043e\u0441\u0441\u0430 \u043f\u043e \u043b\u043e\u0433\u0438\u0442\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\frac{\\partial L}{\\partial z} = \\widehat{y} - y\" alt=\"\\frac{\\partial L}{\\partial z} = \\widehat{y} - y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/324\/324288828c27d865eb9a4ad123aa13f1.svg\" width=\"95\" height=\"38\" data-width=\"11.817\" data-height=\"4.749\" data-vertical-align=\"-1.809\" style=\"width: 11.817ex; height: 4.749ex; vertical-align: -1.809ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/324\/324288828c27d865eb9a4ad123aa13f1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/324\/324288828c27d865eb9a4ad123aa13f1.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043b\u0435\u0436\u0438\u0442 \u0432 \u043f\u0435\u0440\u0432\u0438\u0447\u043d\u043e\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439\/\u0441\u0440\u0435\u0434\u043d\u0438\u0445, \u0440\u0430\u0441\u0441\u0447\u0435\u0442 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043e\u0432 \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u0438\u0442 \u0435\u0433\u043e \u0432 \u0441\u043e\u043f\u0440\u044f\u0436\u0435\u043d\u043d\u043e\u0435 (\u043a\u0430\u043a <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla\\psi(y)\" alt=\"\\nabla\\psi(y)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/caf\/caf6217de31e99e966c3a8cf2088a563.svg\" width=\"50\" height=\"18\" data-width=\"6.226\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.226ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/caf\/caf6217de31e99e966c3a8cf2088a563.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/caf\/caf6217de31e99e966c3a8cf2088a563.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u0417\u0430 \u043e\u0434\u043d\u0443 \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u044e \u0432\u0435\u0441\u0430 \u043c\u043e\u0434\u0435\u043b\u0438 \u043e\u0431\u043d\u043e\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u043d\u0430 \u0438\u0445 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442. \u041f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f, forward \u0438 backward pass \u2013 \u044d\u0442\u043e \u043f\u043e\u043b\u043d\u044b\u0439 \u043a\u0440\u0443\u0433 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0439 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430.<\/p>\n<\/li>\n<\/ol>\n<p>\u0412\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043a \u0437\u0430\u0434\u0430\u0447\u0435 \u0431\u0438\u043d\u0430\u0440\u043d\u043e\u0439 \u043a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u0438 \u0438 \u043f\u043e\u0434\u043e\u0439\u0434\u0435\u043c \u043a \u043d\u0435\u0439 \u0441 \u0434\u0440\u0443\u0433\u043e\u0439 \u0441\u0442\u043e\u0440\u043e\u043d\u044b. \u041f\u0443\u0441\u0442\u044c \u0443 \u043d\u0430\u0441 \u0435\u0441\u0442\u044c \u043b\u0438\u043d\u0435\u0439\u043d\u043e \u0440\u0430\u0437\u0434\u0435\u043b\u0438\u043c\u0430\u044f \u0432\u044b\u0431\u043e\u0440\u043a\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\left( x_{i},y_{i} \\right)\" alt=\"\\left( x_{i},y_{i} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/211\/211ca5cad6b6cbec107c67c17e794831.svg\" width=\"53\" height=\"18\" data-width=\"6.648\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.648ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/211\/211ca5cad6b6cbec107c67c17e794831.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/211\/211ca5cad6b6cbec107c67c17e794831.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0433\u0434\u0435 \u043c\u0435\u0442\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"y_{i} = \\pm 1\" alt=\"y_{i} = \\pm 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a0\/a07\/a070c3a81e144ccc774ebcd77936dacd.svg\" width=\"62\" height=\"18\" data-width=\"7.757\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.757ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a0\/a07\/a070c3a81e144ccc774ebcd77936dacd.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a0\/a07\/a070c3a81e144ccc774ebcd77936dacd.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0418\u0449\u0435\u043c \u0433\u0438\u043f\u0435\u0440\u043f\u043b\u043e\u0441\u043a\u043e\u0441\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"w^{T}x + b = 0\" alt=\"w^{T}x + b = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ab\/abc\/abc1b4ee2dac034e7b13f03e46989633.svg\" width=\"97\" height=\"18\" data-width=\"12.113\" data-height=\"2.203\" data-vertical-align=\"-0.186\" style=\"width: 12.113ex; height: 2.203ex; vertical-align: -0.186ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ab\/abc\/abc1b4ee2dac034e7b13f03e46989633.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ab\/abc\/abc1b4ee2dac034e7b13f03e46989633.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u0437\u0434\u0435\u0441\u044c \u0443\u0436\u0435 \u0443\u0434\u043e\u0431\u043d\u043e \u0440\u0430\u0437\u0434\u0435\u043b\u0438\u0442\u044c \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u044b \u0438 \u0441\u043c\u0435\u0449\u0435\u043d\u0438\u0435), \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0440\u0430\u0437\u0434\u0435\u043b\u0438\u0442 \u043a\u043b\u0430\u0441\u0441\u044b \u0441 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u043c \u0437\u0430\u0437\u043e\u0440\u043e\u043c:<\/p>\n<figure class=\"full-width \"><img decoding=\"async\" src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/9b3\/b14\/f7b\/9b3b14f7b984d438a33a6fba6083c358.jpeg\" alt=\"Support Vector Machines (SVMs): Important Derivations | Towards Data Science\" width=\"561\" height=\"546\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/9b3\/b14\/f7b\/9b3b14f7b984d438a33a6fba6083c358.jpeg 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/9b3\/b14\/f7b\/9b3b14f7b984d438a33a6fba6083c358.jpeg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<div><figcaption>Support Vector Machines (SVMs): Important Derivations | Towards Data Science<\/figcaption><\/div>\n<\/figure>\n<p>\u041f\u0440\u044f\u043c\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\min{\\frac{1}{2}\\left\\| w \\right\\|_{2}^{2}},\\ \\ \u043f\u0440\u0438\\ y_{i}\\left( w^{T}x_{i} + b \\right) \\geq 1\\ \\forall i\" alt=\"\\min{\\frac{1}{2}\\left\\| w \\right\\|_{2}^{2}},\\ \\ \u043f\u0440\u0438\\ y_{i}\\left( w^{T}x_{i} + b \\right) \\geq 1\\ \\forall i\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e0a\/e0a1beb296ddbef470ac624dd9c7715a.svg\" width=\"311\" height=\"37\" data-width=\"38.933\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 38.933ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e0a\/e0a1beb296ddbef470ac624dd9c7715a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e0\/e0a\/e0a1beb296ddbef470ac624dd9c7715a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0427\u0442\u043e\u0431\u044b \u0443\u0447\u0435\u0441\u0442\u044c \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f, \u0432\u0432\u0435\u0434\u0435\u043c \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\alpha_{i} \\geq 0\" alt=\"\\alpha_{i} \\geq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f5f\/f5fedf50a32093eb1f485a340470d122.svg\" width=\"51\" height=\"18\" data-width=\"6.336\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.336ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f5f\/f5fedf50a32093eb1f485a340470d122.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f5f\/f5fedf50a32093eb1f485a340470d122.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"L(w,b,\\alpha) = \\frac{1}{2}\\left\\| w \\right\\|_{2}^{2} - \\sum_{}^{}{\\alpha_{i}\\left( y_{i}\\left( w^{T}x_{i} + b \\right) - 1 \\right)}\" alt=\"L(w,b,\\alpha) = \\frac{1}{2}\\left\\| w \\right\\|_{2}^{2} - \\sum_{}^{}{\\alpha_{i}\\left( y_{i}\\left( w^{T}x_{i} + b \\right) - 1 \\right)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05a\/05ad39dcf0bf3d058bfa2602aaba5b9e.svg\" width=\"389\" height=\"45\" data-width=\"48.569\" data-height=\"5.638\" data-vertical-align=\"-2.253\" style=\"width: 48.569ex; height: 5.638ex; vertical-align: -2.253ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05a\/05ad39dcf0bf3d058bfa2602aaba5b9e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05a\/05ad39dcf0bf3d058bfa2602aaba5b9e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><img decoding=\"async\" class=\"formula\" source=\"\\frac{\\partial L}{\\partial w} = w - \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}} = 0,\\ \\ \\frac{\\partial L}{\\partial b} = - \\sum_{}^{}{\\alpha_{i}y_{i}} = 0\\ \\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ w = \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}},\\ \\ \\sum_{}^{}{\\alpha_{i}y_{i}} = 0\" alt=\"\\frac{\\partial L}{\\partial w} = w - \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}} = 0,\\ \\ \\frac{\\partial L}{\\partial b} = - \\sum_{}^{}{\\alpha_{i}y_{i}} = 0\\ \\ \\ \\\u00a0 \\Rightarrow \\ \\ \\ w = \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}},\\ \\ \\sum_{}^{}{\\alpha_{i}y_{i}} = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/93\/938\/93884c63f995a676baa5b9a9b7aae416.svg\" width=\"674\" height=\"46\" data-width=\"84.27\" data-height=\"5.749\" data-vertical-align=\"-2.309\" style=\"width: 84.27ex; height: 5.749ex; vertical-align: -2.309ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/93\/938\/93884c63f995a676baa5b9a9b7aae416.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/93\/938\/93884c63f995a676baa5b9a9b7aae416.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c \u043e\u0431\u0440\u0430\u0442\u043d\u043e \u0432 \u043b\u0430\u0433\u0440\u0430\u043d\u0436\u0438\u0430\u043d, \u043f\u043e\u0441\u043b\u0435 \u0432\u0441\u0435\u0445 \u0432\u044b\u043a\u043b\u0430\u0434\u043e\u043a \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u0443\u044e \u0437\u0430\u0434\u0430\u0447\u0443:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\max_{\\alpha}L = \\max_{\\alpha}\\left( \\sum_{i}^{}\\alpha_{i} - \\frac{1}{2}\\sum_{i,j}^{}{\\alpha_{i}\\alpha_{j}y_{i}y_{j}x_{i}^{T}x_{j}} \\right),\\ \\ \u043f\u0440\u0438\\ \\sum_{i}^{}{\\alpha_{i}y_{i}} = 0\\ \u0438\\ \\alpha_{i} \\geq 0\" alt=\"\\max_{\\alpha}L = \\max_{\\alpha}\\left( \\sum_{i}^{}\\alpha_{i} - \\frac{1}{2}\\sum_{i,j}^{}{\\alpha_{i}\\alpha_{j}y_{i}y_{j}x_{i}^{T}x_{j}} \\right),\\ \\ \u043f\u0440\u0438\\ \\sum_{i}^{}{\\alpha_{i}y_{i}} = 0\\ \u0438\\ \\alpha_{i} \\geq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4b0\/4b01ebf5844e9a88797d1da2a6a63264.svg\" width=\"583\" height=\"56\" data-width=\"72.903\" data-height=\"6.964\" data-vertical-align=\"-2.916\" style=\"width: 72.903ex; height: 6.964ex; vertical-align: -2.916ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4b0\/4b01ebf5844e9a88797d1da2a6a63264.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4b\/4b0\/4b01ebf5844e9a88797d1da2a6a63264.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0421 \u0442\u043e\u0447\u043a\u0438 \u0437\u0440\u0435\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430, \u043f\u0440\u043e\u0438\u0437\u043e\u0448\u043b\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0437\u043d\u0430\u043a\u043e\u043c\u043e\u0439 \u043d\u0430\u043c \u0441\u0442\u0430\u0446\u0438\u043e\u043d\u0430\u0440\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"f(w) = \\frac{1}{2}\\left\\| w \\right\\|_{2}^{2},\\ \\ f^{*}(\\lambda) = \\sup_{w}\\left( \\lambda^{T}w - \\frac{1}{2}\\left\\| w \\right\\|_{2}^{2} \\right) = \\frac{1}{2}\\left\\| \\lambda \\right\\|_{2}^{2}\" alt=\"f(w) = \\frac{1}{2}\\left\\| w \\right\\|_{2}^{2},\\ \\ f^{*}(\\lambda) = \\sup_{w}\\left( \\lambda^{T}w - \\frac{1}{2}\\left\\| w \\right\\|_{2}^{2} \\right) = \\frac{1}{2}\\left\\| \\lambda \\right\\|_{2}^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e2\/e25\/e25158cf472afae96387bb8a2780d0f5.svg\" width=\"469\" height=\"43\" data-width=\"58.582\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 58.582ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e2\/e25\/e25158cf472afae96387bb8a2780d0f5.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e2\/e25\/e25158cf472afae96387bb8a2780d0f5.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041d\u043e \u0432 \u043d\u0430\u0448\u0435\u0439 \u0437\u0430\u0434\u0430\u0447\u0435 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"y_{i}\\left( w^{T}x_{i} + b \\right) \\geq 1\" alt=\"y_{i}\\left( w^{T}x_{i} + b \\right) \\geq 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/fff\/fffb24a3567c6d14ff8526c6b1a74faa.svg\" width=\"137\" height=\"22\" data-width=\"17.15\" data-height=\"2.808\" data-vertical-align=\"-0.838\" style=\"width: 17.15ex; height: 2.808ex; vertical-align: -0.838ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/fff\/fffb24a3567c6d14ff8526c6b1a74faa.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/ff\/fff\/fffb24a3567c6d14ff8526c6b1a74faa.svg 781w\" loading=\"lazy\" decode=\"async\"\/>\u043b\u0438\u043d\u0435\u0439\u043d\u044b, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u043e\u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\lambda = \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}}\" alt=\"\\lambda = \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d8b\/d8bcf0412bfcfe4c3187cc782bae611c.svg\" width=\"112\" height=\"43\" data-width=\"14.05\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 14.05ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d8b\/d8bcf0412bfcfe4c3187cc782bae611c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d8\/d8b\/d8bcf0412bfcfe4c3187cc782bae611c.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0430 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u043e\u0440\u043e\u0436\u0434\u0430\u0435\u0442 \u0431\u0438\u043b\u0438\u043d\u0435\u0439\u043d\u0443\u044e \u0444\u043e\u0440\u043c\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\sum_{i,j}^{}{\\alpha_{i}\\alpha_{j}y_{i}y_{j}x_{i}^{T}x_{j}}\" alt=\"\\sum_{i,j}^{}{\\alpha_{i}\\alpha_{j}y_{i}y_{j}x_{i}^{T}x_{j}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/eb\/eba\/eba4f6faf47c3b09c5cc475421d5d1be.svg\" width=\"133\" height=\"47\" data-width=\"16.68\" data-height=\"5.834\" data-vertical-align=\"-2.351\" style=\"width: 16.68ex; height: 5.834ex; vertical-align: -2.351ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/eb\/eba\/eba4f6faf47c3b09c5cc475421d5d1be.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/eb\/eba\/eba4f6faf47c3b09c5cc475421d5d1be.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041c\u044b \u043f\u0435\u0440\u0435\u0448\u043b\u0438 \u0438\u0437 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 \u0432\u0435\u0441\u043e\u0432 <img decoding=\"async\" class=\"formula inline\" source=\"w\" alt=\"w\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f1\/f12\/f1290186a5d0b1ceab27f4e77c0c5d68.svg\" width=\"13\" height=\"12\" data-width=\"1.62\" data-height=\"1.027\" data-vertical-align=\"-0.025\" style=\"width: 1.62ex; height: 1.027ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f1\/f12\/f1290186a5d0b1ceab27f4e77c0c5d68.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f1\/f12\/f1290186a5d0b1ceab27f4e77c0c5d68.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439 \u041b\u0430\u0433\u0440\u0430\u043d\u0436\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\alpha\" alt=\"\\alpha\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg\" width=\"12\" height=\"18\" data-width=\"1.448\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.448ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0433\u0434\u0435 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f \u0441\u0442\u0430\u043b\u0438 \u043b\u0438\u043d\u0435\u0439\u043d\u044b\u043c\u0438 (<img decoding=\"async\" class=\"formula inline\" source=\"\\sum_{}^{}{\\alpha_{i}y_{i}} = 0\" alt=\"\\sum_{}^{}{\\alpha_{i}y_{i}} = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0f\/0f4\/0f481c4421d9331fd3c6d0cdb53c4dee.svg\" width=\"95\" height=\"43\" data-width=\"11.828\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 11.828ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0f\/0f4\/0f481c4421d9331fd3c6d0cdb53c4dee.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0f\/0f4\/0f481c4421d9331fd3c6d0cdb53c4dee.svg 781w\" loading=\"lazy\" decode=\"async\"\/>) \u0438 \u043d\u0435\u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u043c\u0438 (<img decoding=\"async\" class=\"formula inline\" source=\"\\alpha_{i} \\geq 0\" alt=\"\\alpha_{i} \\geq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f5f\/f5fedf50a32093eb1f485a340470d122.svg\" width=\"51\" height=\"18\" data-width=\"6.336\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.336ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f5f\/f5fedf50a32093eb1f485a340470d122.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f5\/f5f\/f5fedf50a32093eb1f485a340470d122.svg 781w\" loading=\"lazy\" decode=\"async\"\/>)<em>.<\/em> \u0410 \u043a\u043e\u0433\u0434\u0430 \u043c\u044b \u043d\u0430\u0448\u043b\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\alpha\" alt=\"\\alpha\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg\" width=\"12\" height=\"18\" data-width=\"1.448\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.448ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7b\/7b7\/7b7f9dbfea05c83784f8b85149852f08.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0432\u0435\u0441\u0430 \u0432\u043e\u0441\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u043c \u043f\u043e \u0444\u043e\u0440\u043c\u0443\u043b\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"w = \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}}\" alt=\"w = \\sum_{}^{}{\\alpha_{i}y_{i}x_{i}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/52\/52c\/52c5a272ed7ac196b1fc6edf66f4ae32.svg\" width=\"115\" height=\"43\" data-width=\"14.351\" data-height=\"5.43\" data-vertical-align=\"-2.149\" style=\"width: 14.351ex; height: 5.43ex; vertical-align: -2.149ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/52\/52c\/52c5a272ed7ac196b1fc6edf66f4ae32.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/52\/52c\/52c5a272ed7ac196b1fc6edf66f4ae32.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0430 \u0441\u043c\u0435\u0449\u0435\u043d\u0438\u0435 \u2013 \u0438\u0437 <img decoding=\"async\" class=\"formula inline\" source=\"y_{i}\\left( w^{T}x_{i} + b \\right) = 1\" alt=\"y_{i}\\left( w^{T}x_{i} + b \\right) = 1\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/24\/248\/248574dc330b8d838fac69f4fc26065a.svg\" width=\"137\" height=\"22\" data-width=\"17.15\" data-height=\"2.808\" data-vertical-align=\"-0.838\" style=\"width: 17.15ex; height: 2.808ex; vertical-align: -0.838ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/24\/248\/248574dc330b8d838fac69f4fc26065a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/24\/248\/248574dc330b8d838fac69f4fc26065a.svg 781w\" loading=\"lazy\" decode=\"async\"\/><em>.<\/em> \u0415\u0441\u043b\u0438 \u0434\u0430\u043d\u043d\u044b\u0435 \u043d\u0435 \u0440\u0430\u0437\u0434\u0435\u043b\u0438\u043c\u044b \u043b\u0438\u043d\u0435\u0439\u043d\u043e, \u0442\u043e \u0432\u0432\u043e\u0434\u044f\u0442\u0441\u044f \u0448\u0442\u0440\u0430\u0444 \u0437\u0430 \u043d\u0430\u0445\u043b\u0435\u0441\u0442 \u0441 \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u043e\u043c.<\/p>\n<p>\u0411\u043e\u043b\u0435\u0435 \u0442\u043e\u0433\u043e, \u0432 \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0435 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"x_{j}\" alt=\"x_{j}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4b\/f4bb95e27d8505366199bb81f2b75b6f.svg\" width=\"17\" height=\"13\" data-width=\"2.141\" data-height=\"1.666\" data-vertical-align=\"-0.666\" style=\"width: 2.141ex; height: 1.666ex; vertical-align: -0.666ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4b\/f4bb95e27d8505366199bb81f2b75b6f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4b\/f4bb95e27d8505366199bb81f2b75b6f.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u0445\u043e\u0434\u044f\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u0447\u0435\u0440\u0435\u0437 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"x_{i}^{T}x_{j}\" alt=\"x_{i}^{T}x_{j}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c29\/c294a7c1b08ff06468af4298ded38126.svg\" width=\"38\" height=\"21\" data-width=\"4.749\" data-height=\"2.683\" data-vertical-align=\"-0.666\" style=\"width: 4.749ex; height: 2.683ex; vertical-align: -0.666ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c29\/c294a7c1b08ff06468af4298ded38126.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c29\/c294a7c1b08ff06468af4298ded38126.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041c\u043e\u0436\u043d\u043e \u0437\u0430\u043c\u0435\u043d\u0438\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"x_{i}^{T}x_{j}\" alt=\"x_{i}^{T}x_{j}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c29\/c294a7c1b08ff06468af4298ded38126.svg\" width=\"38\" height=\"21\" data-width=\"4.749\" data-height=\"2.683\" data-vertical-align=\"-0.666\" style=\"width: 4.749ex; height: 2.683ex; vertical-align: -0.666ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c29\/c294a7c1b08ff06468af4298ded38126.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c2\/c29\/c294a7c1b08ff06468af4298ded38126.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u0443\u044e \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u0443\u044e \u044f\u0434\u0435\u0440\u043d\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u044e <img decoding=\"async\" class=\"formula inline\" source=\"K\\left( x_{i},x_{j} \\right) = \\psi\\left( x_{i} \\right)^{T}\\psi\\left( x_{j} \\right)\" alt=\"K\\left( x_{i},x_{j} \\right) = \\psi\\left( x_{i} \\right)^{T}\\psi\\left( x_{j} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8a\/f8af12f6c098aeae274d636ef54fae7e.svg\" width=\"197\" height=\"23\" data-width=\"24.679\" data-height=\"2.828\" data-vertical-align=\"-0.848\" style=\"width: 24.679ex; height: 2.828ex; vertical-align: -0.848ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8a\/f8af12f6c098aeae274d636ef54fae7e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f8a\/f8af12f6c098aeae274d636ef54fae7e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0437\u0430\u0434\u0430\u0435\u0442 \u043c\u0435\u0442\u0440\u0438\u043a\u0443 \u0438 \u0434\u0438\u0432\u0435\u0440\u0433\u0435\u043d\u0446\u0438\u044e \u0411\u0440\u0435\u0433\u043c\u0430\u043d\u0430 \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u043e\u0432:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\max_{\\alpha}\\left( \\sum_{i}^{}\\alpha_{i} - \\frac{1}{2}\\sum_{i,j}^{}{\\alpha_{i}\\alpha_{j}y_{i}y_{j}K\\left( x_{i},x_{j} \\right)} \\right)\" alt=\"\\max_{\\alpha}\\left( \\sum_{i}^{}\\alpha_{i} - \\frac{1}{2}\\sum_{i,j}^{}{\\alpha_{i}\\alpha_{j}y_{i}y_{j}K\\left( x_{i},x_{j} \\right)} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c0f\/c0fd19ad42165e006ec8ab8b8abcd4f7.svg\" width=\"324\" height=\"56\" data-width=\"40.532\" data-height=\"6.964\" data-vertical-align=\"-2.916\" style=\"width: 40.532ex; height: 6.964ex; vertical-align: -2.916ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c0f\/c0fd19ad42165e006ec8ab8b8abcd4f7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c0\/c0f\/c0fd19ad42165e006ec8ab8b8abcd4f7.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e\u0442 \u043c\u0435\u0442\u043e\u0434 \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043c\u0435\u0442\u043e\u0434\u043e\u043c \u043e\u043f\u043e\u0440\u043d\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 (SVM). \u041c\u0435\u0442\u043e\u0434 \u043d\u0430\u0433\u043b\u044f\u0434\u043d\u043e \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442 \u043f\u043e\u043b\u044c\u0437\u0443 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430: \u043d\u0430\u043c \u0443\u0434\u0430\u043b\u043e\u0441\u044c \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u043a \u0437\u0430\u0434\u0430\u0447\u0435 \u0432 \u0442\u0430\u043a\u0438\u0445 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430\u0445, \u0447\u0442\u043e \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u043b\u0430 \u0437\u0430\u0432\u0438\u0441\u0435\u0442\u044c \u043e\u0442 \u0447\u0438\u0441\u043b\u0430 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u043e\u0432 (\u0442\u0435\u043f\u0435\u0440\u044c \u2013 \u0442\u043e\u043b\u044c\u043a\u043e \u043e\u0442 \u0447\u0438\u0441\u043b\u0430 \u043e\u0431\u044a\u0435\u043a\u0442\u043e\u0432), \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f \u0441\u0442\u0430\u043b\u0438 \u043f\u0440\u043e\u0441\u0442\u044b\u043c\u0438 (\u043b\u0438\u043d\u0435\u0439\u043d\u044b\u043c\u0438), \u0430 \u0432\u0441\u044f \u043d\u0435\u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0441\u0442\u044c \u0443\u0448\u043b\u0430 \u0432 \u044f\u0434\u0440\u0430 (\u0441\u043a\u0430\u043b\u044f\u0440\u043d\u044b\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u043e\u0432)<\/p>\n<p>\u0413\u0435\u043d\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u044b\u0435 \u043c\u043e\u0434\u0435\u043b\u0438, ELBO<\/p>\n<p>\u0412\u0435\u0440\u043d\u0435\u043c\u0441\u044f \u043a \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0439 \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u0438: \u0447\u0442\u043e \u0442\u0430\u043c \u043d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u043b\u043e? \u0412 \u043d\u0430\u0447\u0430\u043b\u0435 \u043c\u044b \u0441\u0434\u0435\u043b\u0430\u043b\u0438 \u0440\u044f\u0434 \u0430\u043f\u0440\u0438\u043e\u0440\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0439:<\/p>\n<ol>\n<li>\n<p>\u043e \u0434\u0430\u043d\u043d\u044b\u0445 &#8212; \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u044f \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u044b \u0438 \u043f\u043e\u0440\u043e\u0436\u0434\u0435\u043d\u044b \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u043c\u043e\u0434\u0435\u043b\u044c\u044e<\/p>\n<\/li>\n<li>\n<p>\u043e\u0431 \u044d\u0442\u043e\u0439 \u043c\u043e\u0434\u0435\u043b\u0438 &#8212; \u0442\u0430\u0440\u0433\u0435\u0442 \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u043e\u0442 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u043e\u0432 \u043b\u0438\u043d\u0435\u0439\u043d\u043e (\u0441 \u0432\u0435\u0441\u0430\u043c\u0438 \u0438 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u043c \u0448\u0443\u043c\u043e\u043c)<\/p>\n<\/li>\n<li>\n<p>\u043e\u0431 \u044d\u0442\u043e\u043c \u0448\u0443\u043c\u0435 \u2013 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0441\u0442\u044c \u0438 \u0433\u043e\u043c\u043e\u0441\u043a\u0435\u0434\u0430\u0441\u0442\u0438\u0447\u043d\u043e\u0441\u0442\u044c, <img decoding=\"async\" class=\"formula inline\" source=\"\\varepsilon\\ \\sim\\ N\\left( 0,\\sigma^{2}I \\right)\" alt=\"\\varepsilon\\ \\sim\\ N\\left( 0,\\sigma^{2}I \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f0\/f0b\/f0b250a7e89e809c986049e8bfc2b383.svg\" width=\"122\" height=\"22\" data-width=\"15.218\" data-height=\"2.791\" data-vertical-align=\"-0.83\" style=\"width: 15.218ex; height: 2.791ex; vertical-align: -0.83ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f0\/f0b\/f0b250a7e89e809c986049e8bfc2b383.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f0\/f0b\/f0b250a7e89e809c986049e8bfc2b383.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p>\u0438 \u043e\u0431 \u044d\u0442\u0438\u0445 \u0432\u0435\u0441\u0430\u0445 &#8212; \u0442\u043e\u0436\u0435 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0441\u0442\u044c \u0438 \u0433\u043e\u043c\u043e\u0441\u043a\u0435\u0434\u0430\u0441\u0442\u0438\u0447\u043d\u043e\u0441\u0442\u044c, <img decoding=\"async\" class=\"formula inline\" source=\"\\theta_{0}\\ \\sim\\ N\\left( 0,\\sigma_{0}^{2}I \\right)\" alt=\"\\theta_{0}\\ \\sim\\ N\\left( 0,\\sigma_{0}^{2}I \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c4\/c44\/c4459281580cd91eb09b7064858f1b3e.svg\" width=\"130\" height=\"22\" data-width=\"16.213\" data-height=\"2.791\" data-vertical-align=\"-0.83\" style=\"width: 16.213ex; height: 2.791ex; vertical-align: -0.83ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c4\/c44\/c4459281580cd91eb09b7064858f1b3e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c4\/c44\/c4459281580cd91eb09b7064858f1b3e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<\/ol>\n<p>\u041f\u043e\u0442\u043e\u043c \u043c\u044b \u0437\u0430\u0434\u0443\u043c\u0430\u043b\u0438\u0441\u044c: \u043a\u0430\u043a\u0430\u044f \u043c\u043e\u0434\u0435\u043b\u044c <img decoding=\"async\" class=\"formula inline\" source=\"f_{\\widehat{\\theta}}(x)\" alt=\"f_{\\widehat{\\theta}}(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdf\/fdfeb4a0bd03fb600b21cbbcd7a62f00.svg\" width=\"41\" height=\"21\" data-width=\"5.101\" data-height=\"2.629\" data-vertical-align=\"-0.749\" style=\"width: 5.101ex; height: 2.629ex; vertical-align: -0.749ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdf\/fdfeb4a0bd03fb600b21cbbcd7a62f00.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fd\/fdf\/fdfeb4a0bd03fb600b21cbbcd7a62f00.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432 \u044d\u0442\u0438\u0445 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u044f\u0445 \u043d\u0430\u0438\u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u043d\u043e \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442 \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u044f? \u041e\u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\widehat{\\theta} = \\arg{\\max_{\\theta}{P\\left( Y\\ |\\ X,\\theta \\right)}} = \\left( X^{T}X + \\frac{\\sigma^{2}}{\\sigma_{0}^{2}}I \\right)^{- 1}X^{T}Y\" alt=\"\\widehat{\\theta} = \\arg{\\max_{\\theta}{P\\left( Y\\ |\\ X,\\theta \\right)}} = \\left( X^{T}X + \\frac{\\sigma^{2}}{\\sigma_{0}^{2}}I \\right)^{- 1}X^{T}Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/369\/3698aa27f0ae0bc1d4f998871a553d18.svg\" width=\"399\" height=\"58\" data-width=\"49.892\" data-height=\"7.233\" data-vertical-align=\"-3.051\" style=\"width: 49.892ex; height: 7.233ex; vertical-align: -3.051ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/369\/3698aa27f0ae0bc1d4f998871a553d18.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/36\/369\/3698aa27f0ae0bc1d4f998871a553d18.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u043d\u0430\u044f \u043c\u043e\u0434\u0435\u043b\u044c \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0434\u0438\u0441\u043a\u0440\u0438\u043c\u0438\u043d\u0430\u0442\u0438\u0432\u043d\u043e\u0439: \u044d\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e \u043f\u043e \u0434\u0430\u043d\u043d\u044b\u043c \u043d\u0430\u043c \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0430\u043c <img decoding=\"async\" class=\"formula inline\" source=\"X\" alt=\"X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg\" width=\"15\" height=\"12\" data-width=\"1.928\" data-height=\"1.545\" style=\"width: 1.928ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043c\u044b \u0443\u043c\u0435\u0435\u043c \u0432\u043e\u0441\u0441\u0442\u0430\u043d\u0430\u0432\u043b\u0438\u0432\u0430\u0442\u044c \u043d\u0430\u0438\u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u043d\u044b\u0439 \u0442\u0430\u0440\u0433\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"Y\" alt=\"Y\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg\" width=\"14\" height=\"12\" data-width=\"1.726\" data-height=\"1.545\" style=\"width: 1.726ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/57\/57c\/57cec4137b614c87cb4e24a3d003a3e0.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e \u0441\u0430\u043c\u043e\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0438 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u043e\u0432 <img decoding=\"async\" class=\"formula inline\" source=\"X\" alt=\"X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg\" width=\"15\" height=\"12\" data-width=\"1.928\" data-height=\"1.545\" style=\"width: 1.928ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043c\u044b \u043d\u0438\u0447\u0435\u0433\u043e \u043d\u0435 \u0437\u043d\u0430\u0435\u043c. \u041f\u043e\u043a\u0440\u0443\u0442\u0438\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0443\u0441\u043b\u043e\u0432\u043d\u043e\u0439 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"P\\left( Y\\ |\\ X,\\theta \\right) = \\frac{P\\left( X,Y\\ |\\ \\theta \\right)}{P\\left( X\\ |\\ \\theta \\right)}\\\u00a0 \\propto \\ P\\left( X,Y\\ |\\ \\theta \\right)\" alt=\"P\\left( Y\\ |\\ X,\\theta \\right) = \\frac{P\\left( X,Y\\ |\\ \\theta \\right)}{P\\left( X\\ |\\ \\theta \\right)}\\\u00a0 \\propto \\ P\\left( X,Y\\ |\\ \\theta \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8c7\/8c7c86f1c9c7aa3d18916ea84590caf3.svg\" width=\"337\" height=\"44\" data-width=\"42.113\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 42.113ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8c7\/8c7c86f1c9c7aa3d18916ea84590caf3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8c7\/8c7c86f1c9c7aa3d18916ea84590caf3.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u043c\u044b \u0445\u043e\u0442\u0438\u043c \u0441\u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0440\u0435\u0430\u043b\u0438\u0441\u0442\u0438\u0447\u043d\u0443\u044e \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u043a\u043e\u0442\u043e\u0432 &#8212; \u043d\u0430\u043c \u043d\u0443\u0436\u043d\u0430 \u043c\u043e\u0434\u0435\u043b\u044c <img decoding=\"async\" class=\"formula inline\" source=\"P\\left( img,\\ \u043a\u043e\u0442\\ |\\ \\theta \\right)\" alt=\"P\\left( img,\\ \u043a\u043e\u0442\\ |\\ \\theta \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/937\/112\/9a2\/9371129a22857a41614d17e1e91ae739.svg\" width=\"134\" height=\"19\" data-width=\"16.148\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 16.148ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/937\/112\/9a2\/9371129a22857a41614d17e1e91ae739.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/937\/112\/9a2\/9371129a22857a41614d17e1e91ae739.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0414\u0438\u0441\u043a\u0440\u0438\u043c\u0438\u043d\u0430\u0442\u0438\u0432\u043d\u044b\u0435 \u043c\u043e\u0434\u0435\u043b\u0438 \u043d\u0430\u043c \u043d\u0435 \u043f\u043e\u043c\u043e\u0433\u0443\u0442: \u043e\u043d\u0438 \u043c\u043e\u0433\u0443\u0442 \u043b\u0438\u0448\u044c \u0441\u043e\u043f\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0442\u044c \u0434\u0430\u043d\u043d\u043e\u0439 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0435 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0438\u043b\u0438 \u043c\u0435\u0442\u043a\u0443 \u043a\u043b\u0430\u0441\u0441\u0430: <img decoding=\"async\" class=\"formula inline\" source=\"P\\left( \u043a\u043e\u0442\\ |\\ img,\\ \\theta \\right)\" alt=\"P\\left( \u043a\u043e\u0442\\ |\\ img,\\ \\theta \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/1f0\/a8b\/fb7\/1f0a8bfb725d283c3ce2d5096ab71d33.svg\" width=\"134\" height=\"19\" data-width=\"16.148\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 16.148ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/1f0\/a8b\/fb7\/1f0a8bfb725d283c3ce2d5096ab71d33.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/1f0\/a8b\/fb7\/1f0a8bfb725d283c3ce2d5096ab71d33.svg 781w\" loading=\"lazy\" decode=\"async\"\/> &#8212; \u00ab\u043d\u0430 \u0434\u0430\u043d\u043d\u043e\u0439 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0435 \u043a\u043e\u0442\u00bb. \u0423\u043c\u0435\u044f \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043a\u043e\u0442\u043e\u0432, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043b\u0435\u0433\u043a\u043e \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u0438\u0445 \u043a\u043b\u0430\u0441\u0441\u0438\u0444\u0438\u043a\u0430\u0442\u043e\u0440: <img decoding=\"async\" class=\"formula inline\" source=\"P\\left( \u043a\u043e\u0442\\ |\\ img,\\ \\theta \\right) \\propto P\\left( img,\u043a\u043e\u0442\\ |\\ \\theta \\right)\" alt=\"P\\left( \u043a\u043e\u0442\\ |\\ img,\\ \\theta \\right) \\propto P\\left( img,\u043a\u043e\u0442\\ |\\ \\theta \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/1b5\/5fe\/00d\/1b55fe00d088b10e832dfba89e7b332f.svg\" width=\"288\" height=\"19\" data-width=\"34.747\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 34.747ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/1b5\/5fe\/00d\/1b55fe00d088b10e832dfba89e7b332f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/upload_files\/1b5\/5fe\/00d\/1b55fe00d088b10e832dfba89e7b332f.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041d\u043e \u0432\u043e\u0442 \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u0430 \u043f\u0440\u0438\u043d\u0446\u0438\u043f\u0438\u0430\u043b\u044c\u043d\u043e \u0441\u043b\u043e\u0436\u043d\u0435\u0435.<\/p>\n<p>\u041c\u043e\u0434\u0435\u043b\u044c, \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043c\u044b \u0445\u043e\u0442\u0438\u043c, \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u043e\u0439. \u0412 \u043e\u0442\u043b\u0438\u0447\u0438\u0435 \u043e\u0442 \u0434\u0438\u0441\u043a\u0440\u0438\u043c\u0438\u043d\u0430\u0442\u0438\u0432\u043d\u044b\u0445, \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0443\u044e\u0449\u0438\u0445 \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440, \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u0430\u044f \u043c\u043e\u0434\u0435\u043b\u044c \u043f\u043e\u0437\u0432\u043e\u043b\u0438\u0442 \u043d\u0430\u043c \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0441\u043e\u0432\u043c\u0435\u0441\u0442\u043d\u043e\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438 \u0440\u0435\u0448\u0430\u0442\u044c \u0441\u043e\u0432\u0435\u0440\u0448\u0435\u043d\u043d\u043e \u043d\u043e\u0432\u044b\u0439 \u043a\u043b\u0430\u0441\u0441 \u0437\u0430\u0434\u0430\u0447. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u043f\u0440\u0438\u0439\u0442\u0438 \u043a \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u044b\u043c \u043c\u043e\u0434\u0435\u043b\u044f\u043c \u043d\u0430\u043c \u043f\u043e\u043c\u043e\u0436\u0435\u0442 \u0438\u043c\u0435\u043d\u043d\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430.<\/p>\n<p>\u0412\u0441\u0435 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"x\" alt=\"x\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg\" width=\"12\" height=\"12\" data-width=\"1.294\" data-height=\"1.025\" data-vertical-align=\"-0.025\" style=\"width: 1.294ex; height: 1.025ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/9d\/9dd\/9dd4e461268c8034f5c8564e155c67a6.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043b\u0435\u0436\u0430\u0442 \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 3D-\u043c\u0430\u0442\u0440\u0438\u0446 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438 height x width x rgb. \u0411\u043e\u043b\u044c\u0448\u0438\u043d\u0441\u0442\u0432\u043e \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0430 \u0432\u044b\u0433\u043b\u044f\u0434\u044f\u0442 \u043a\u0430\u043a \u0448\u0443\u043c\u043d\u044b\u0439 \u044d\u043a\u0440\u0430\u043d \u0442\u0435\u043b\u0435\u0432\u0438\u0437\u043e\u0440\u0430, \u043d\u043e \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043f\u043e\u0434\u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0438\u0437 \u043d\u0438\u0445 \u2013 \u044d\u0442\u043e \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0438 \u043a\u043e\u0442\u043e\u0432. \u041a\u0430\u0440\u0442\u0438\u043d\u043a\u0438 \u0438\u043c\u0435\u043d\u043d\u043e \u043a\u043e\u0442\u043e\u0432 \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u0441\u043b\u043e\u0436\u043d\u044b\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c, \u0437\u0430\u0432\u0438\u0441\u044f\u0449\u0435\u043c \u043e\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"z\" alt=\"z\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbade9e36a3f36d3d676c1b808451dd7.svg\" width=\"12\" height=\"12\" data-width=\"1.052\" data-height=\"1.025\" data-vertical-align=\"-0.025\" style=\"width: 1.052ex; height: 1.025ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbade9e36a3f36d3d676c1b808451dd7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbade9e36a3f36d3d676c1b808451dd7.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e \u0441\u043a\u0440\u044b\u0442\u044b\u0435 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u044b\u0435 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b (\u043b\u0430\u0442\u0435\u043d\u0442\u044b), \u043a\u043e\u0434\u0438\u0440\u0443\u044e\u0449\u0438\u0435 \u201c\u0441\u043c\u044b\u0441\u043b\u201d \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0438. \u041d\u0430\u0443\u0447\u0438\u043c\u0441\u044f \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043b\u0430\u0442\u0435\u043d\u0442\u044b <img decoding=\"async\" class=\"formula inline\" source=\"z\" alt=\"z\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbade9e36a3f36d3d676c1b808451dd7.svg\" width=\"12\" height=\"12\" data-width=\"1.052\" data-height=\"1.025\" data-vertical-align=\"-0.025\" style=\"width: 1.052ex; height: 1.025ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbade9e36a3f36d3d676c1b808451dd7.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/fb\/fba\/fbade9e36a3f36d3d676c1b808451dd7.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u0432\u043e\u0441\u0441\u0442\u0430\u043d\u0430\u0432\u043b\u0438\u0432\u0430\u0442\u044c \u0438\u0437 \u043d\u0438\u0445 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0438 \u043a\u043e\u0442\u043e\u0432 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043c\u043e\u0434\u0435\u043b\u0438 \u0441 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u043c\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u0411\u0430\u0439\u0435\u0441\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"p_{\\theta}\\left( z\\ |\\ x \\right) = \\frac{p_{\\theta}(x,z)}{p_{\\theta}(x)} = \\frac{p_{\\theta}\\left( x\\ |\\ z \\right)q(z)}{\\int_{}^{}{p_{\\theta}\\left( x\\ |\\ z \\right)q(z)dz}} = \\frac{p_{\\theta}(x,z)}{\\mathbb{E}_{z\\ \\sim\\ q(z)}\\left\\lbrack p_{\\theta}\\left( x\\ |\\ z \\right) \\right\\rbrack}\" alt=\"p_{\\theta}\\left( z\\ |\\ x \\right) = \\frac{p_{\\theta}(x,z)}{p_{\\theta}(x)} = \\frac{p_{\\theta}\\left( x\\ |\\ z \\right)q(z)}{\\int_{}^{}{p_{\\theta}\\left( x\\ |\\ z \\right)q(z)dz}} = \\frac{p_{\\theta}(x,z)}{\\mathbb{E}_{z\\ \\sim\\ q(z)}\\left\\lbrack p_{\\theta}\\left( x\\ |\\ z \\right) \\right\\rbrack}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea3\/ea366fbff940f489a062ff7b68a899cf.svg\" width=\"489\" height=\"46\" data-width=\"61.185\" data-height=\"5.806\" data-vertical-align=\"-2.338\" style=\"width: 61.185ex; height: 5.806ex; vertical-align: -2.338ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea3\/ea366fbff940f489a062ff7b68a899cf.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/ea\/ea3\/ea366fbff940f489a062ff7b68a899cf.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u043e\u0434\u0435\u043b\u044c <img decoding=\"async\" class=\"formula inline\" source=\"p_{\\theta}\\left( x\\ |\\ z \\right)\" alt=\"p_{\\theta}\\left( x\\ |\\ z \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d16\/d169f070118445c3f80e0f5a97583259.svg\" width=\"67\" height=\"18\" data-width=\"8.32\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.32ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d16\/d169f070118445c3f80e0f5a97583259.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/d1\/d16\/d169f070118445c3f80e0f5a97583259.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0434\u0435\u043a\u043e\u0434\u0435\u0440, <img decoding=\"async\" class=\"formula inline\" source=\"p_{\\theta}\\left( z\\ |\\ x \\right)\" alt=\"p_{\\theta}\\left( z\\ |\\ x \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/25f\/25f4337073e18a4340a0b0312ac1175a.svg\" width=\"67\" height=\"18\" data-width=\"8.32\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 8.32ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/25f\/25f4337073e18a4340a0b0312ac1175a.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/25f\/25f4337073e18a4340a0b0312ac1175a.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u043d\u043a\u043e\u0434\u0435\u0440, <img decoding=\"async\" class=\"formula inline\" source=\"q(z)\" alt=\"q(z)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a5\/a58\/a5827c143f7d49ac84e4a10aac2b490c.svg\" width=\"31\" height=\"18\" data-width=\"3.853\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.853ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a5\/a58\/a5827c143f7d49ac84e4a10aac2b490c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a5\/a58\/a5827c143f7d49ac84e4a10aac2b490c.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0430\u043f\u0440\u0438\u043e\u0440 \u043d\u0430 \u043b\u0430\u0442\u0435\u043d\u0442\u044b (\u043d\u0435 \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u043e\u0442 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u043e\u0432 \u043c\u043e\u0434\u0435\u043b\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta\" alt=\"\\theta\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg\" width=\"12\" height=\"18\" data-width=\"1.061\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.061ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/25\/255\/2554a2bb846cffd697389e5dc8912759.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c, \u0447\u0442\u043e \u0437\u043d\u0430\u043c\u0435\u043d\u0430\u0442\u0435\u043b\u044c \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0411\u0430\u0439\u0435\u0441\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"p_{\\theta}(x)\" alt=\"p_{\\theta}(x)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4c\/f4c49fc5b507afef5846a8b0800ea79e.svg\" width=\"41\" height=\"18\" data-width=\"5.13\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 5.13ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4c\/f4c49fc5b507afef5846a8b0800ea79e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f4\/f4c\/f4c49fc5b507afef5846a8b0800ea79e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f evidence: \u0432 \u043d\u0430\u0448\u0435\u043c \u043f\u0440\u0438\u043c\u0435\u0440\u0435 \u0441 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0430\u043c\u0438 \u2013 \u044d\u0442\u043e \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u043e\u0440 \u0432\u0441\u0435\u0445 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u044b\u0445 \u043a\u0430\u0440\u0442\u0438\u043d\u043e\u043a, \u043f\u043e\u0445\u043e\u0436\u0438\u0445 \u043d\u0430 \u043d\u0430\u0448 \u0434\u0430\u0442\u0430\u0441\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"X\" alt=\"X\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg\" width=\"15\" height=\"12\" data-width=\"1.928\" data-height=\"1.545\" style=\"width: 1.928ex; height: 1.545ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/02\/021\/02129bb861061d1a052c592e2dc6b383.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041c\u044b \u0445\u043e\u0442\u0438\u043c, \u0447\u0442\u043e\u0431\u044b \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0438 \u0431\u044b\u043b\u0438 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u043d\u044b\u043c\u0438, \u0442.\u0435. \u0438\u0449\u0435\u043c:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\max_{\\theta}{\\ln{p_{\\theta}(x)}} = \\max_{\\theta}{\\ln{\\int_{}^{}{p_{\\theta}(x,z)dz}}} = \\max_{\\theta}{\\ln{\\int_{}^{}{e^{\\ln{p_{\\theta}(x,z)}}dz}}}\" alt=\"\\max_{\\theta}{\\ln{p_{\\theta}(x)}} = \\max_{\\theta}{\\ln{\\int_{}^{}{p_{\\theta}(x,z)dz}}} = \\max_{\\theta}{\\ln{\\int_{}^{}{e^{\\ln{p_{\\theta}(x,z)}}dz}}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5c\/5cd\/5cd32dc53cb43c22e5f90282d130790f.svg\" width=\"454\" height=\"41\" data-width=\"56.739\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 56.739ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5c\/5cd\/5cd32dc53cb43c22e5f90282d130790f.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5c\/5cd\/5cd32dc53cb43c22e5f90282d130790f.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u042d\u0442\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 &#8212; \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u0432\u0448\u0438\u0439\u0441\u044f \u043d\u0430\u043c \u043c\u043d\u043e\u0433\u043e \u0440\u0430\u0437 logsumexp. \u041c\u044b \u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u0437\u0430\u0434\u0430\u0447\u0430 \u043f\u043e\u0438\u0441\u043a\u0430 \u041e\u041c\u041f \u2013 \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u0430\u044f \u043a \u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u0438 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438, \u0430 logsumexp \u2013 \u044d\u0442\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043e\u0442 \u043d\u0435\u0433\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\int_{}^{}{q(z)\\ln{q(z)dz}}\" alt=\"\\int_{}^{}{q(z)\\ln{q(z)dz}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5f\/5fd\/5fdc499723c224a65686e23b5f984554.svg\" width=\"123\" height=\"41\" data-width=\"15.433\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 15.433ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5f\/5fd\/5fdc499723c224a65686e23b5f984554.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/5\/5f\/5fd\/5fdc499723c224a65686e23b5f984554.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e\u0441\u0442\u0430\u043b\u043e\u0441\u044c \u0442\u043e\u043b\u044c\u043a\u043e \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\ln{p_{\\theta}(x,z)}\" alt=\"\\ln{p_{\\theta}(x,z)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/98\/98a\/98a15911bd55dbce864c27656463b734.svg\" width=\"76\" height=\"18\" data-width=\"9.452\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 9.452ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/98\/98a\/98a15911bd55dbce864c27656463b734.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/98\/98a\/98a15911bd55dbce864c27656463b734.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432\u043c\u0435\u0441\u0442\u043e \u0434\u0432\u043e\u0439\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\ln{p_{\\theta}(x)} = \\max_{q}\\left( \\int_{}^{}{q(z)\\ln{p_{\\theta}(x,z)}dz} - \\int_{}^{}{q(z)\\ln{q(z)}dz} \\right) = \\max_{q}\\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}(x,z)}{q(z)}dz}\" alt=\"\\ln{p_{\\theta}(x)} = \\max_{q}\\left( \\int_{}^{}{q(z)\\ln{p_{\\theta}(x,z)}dz} - \\int_{}^{}{q(z)\\ln{q(z)}dz} \\right) = \\max_{q}\\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}(x,z)}{q(z)}dz}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3b\/3b7\/3b789115a47c302b6c114908403deef0.svg\" width=\"661\" height=\"44\" data-width=\"82.665\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 82.665ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3b\/3b7\/3b789115a47c302b6c114908403deef0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/3b\/3b7\/3b789115a47c302b6c114908403deef0.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0412\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043f\u043e\u0434 \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c\u043e\u043c \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f ELBO \u2013 Evidence Lower BOund. \u041d\u0430\u0437\u0432\u0430\u043d\u0438\u0435 \u043e\u0442\u0441\u044b\u043b\u0430\u0435\u0442 \u043d\u0430\u0441 \u043a \u0440\u0430\u0437\u0433\u043e\u0432\u043e\u0440\u0443 \u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 &#8212; \u043a\u0430\u043a \u043f\u043e\u0438\u0441\u043a\u0443 \u043d\u0430\u0438\u0431\u043e\u043b\u0435\u0435 \u00ab\u0442\u0443\u0433\u043e\u0439\u00bb \u043d\u0438\u0436\u043d\u0438\u0439 \u043e\u0446\u0435\u043d\u043a\u0438 \u043d\u0430 \u0432\u044b\u043f\u0443\u043a\u043b\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u044e. \u0411\u043e\u043b\u0435\u0435 \u0442\u043e\u0433\u043e, \u043f\u043e\u0434\u043e\u0433\u043d\u0430\u0432 \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043a \u0437\u043d\u0430\u043a\u043e\u043c\u043e\u043c\u0443 \u0432\u0438\u0434\u0443 logsumexp, \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0438 \u043d\u0430\u0433\u043b\u044f\u0434\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0434\u043b\u044f \u043b\u0435\u0436\u0430\u043d\u0434\u0440\u043e\u0432\u0430 \u0437\u0430\u0437\u043e\u0440\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"ELBO\\lbrack q\\rbrack = \\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}(x,z)}{q(z)}dz} = \\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}\\left( z\\ |\\ x \\right)p_{\\theta}(x)}{q(z)}dz} = \\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}\\left( z\\ |\\ x \\right)}{q(z)}dz} + \\int_{}^{}{q(z)\\ln{p_{\\theta}(x)}dz} = \\left( \\int_{}^{}{q(z)dz} \\right)\\ln{p_{\\theta}(x)} - \\int_{}^{}{q(z)\\ln\\frac{q(z)}{p_{\\theta}\\left( z\\ |\\ x \\right)}dz} = \\ln{p_{\\theta}(x)} - KL(q(z)\\ ||\\ p_{\\theta}\\left( z\\ |\\ x \\right))\" alt=\"ELBO\\lbrack q\\rbrack = \\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}(x,z)}{q(z)}dz} = \\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}\\left( z\\ |\\ x \\right)p_{\\theta}(x)}{q(z)}dz} = \\int_{}^{}{q(z)\\ln\\frac{p_{\\theta}\\left( z\\ |\\ x \\right)}{q(z)}dz} + \\int_{}^{}{q(z)\\ln{p_{\\theta}(x)}dz} = \\left( \\int_{}^{}{q(z)dz} \\right)\\ln{p_{\\theta}(x)} - \\int_{}^{}{q(z)\\ln\\frac{q(z)}{p_{\\theta}\\left( z\\ |\\ x \\right)}dz} = \\ln{p_{\\theta}(x)} - KL(q(z)\\ ||\\ p_{\\theta}\\left( z\\ |\\ x \\right))\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d5\/6d5c028e3dd93f3806332d838bc81111.svg\" width=\"1472\" height=\"44\" data-width=\"183.994\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 183.994ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d5\/6d5c028e3dd93f3806332d838bc81111.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6d\/6d5\/6d5c028e3dd93f3806332d838bc81111.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <img decoding=\"async\" class=\"formula inline\" source=\"\\ln{p_{\\theta}(x)}\" alt=\"\\ln{p_{\\theta}(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a49265f4d9238fd33fd0ea32b1a04e6d.svg\" width=\"59\" height=\"18\" data-width=\"7.394\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.394ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a49265f4d9238fd33fd0ea32b1a04e6d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a49265f4d9238fd33fd0ea32b1a04e6d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043d\u0435 \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u043e\u0442 q, \u0442\u043e \u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u044f <img decoding=\"async\" class=\"formula inline\" source=\"ELBO\\lbrack q\\rbrack\" alt=\"ELBO\\lbrack q\\rbrack\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dd\/ddc\/ddc1eb93cb588848fb276a329049e915.svg\" width=\"72\" height=\"18\" data-width=\"9.011\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 9.011ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dd\/ddc\/ddc1eb93cb588848fb276a329049e915.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/d\/dd\/ddc\/ddc1eb93cb588848fb276a329049e915.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043c\u044b \u043c\u0438\u043d\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u043c KL-\u0434\u0438\u0432\u0435\u0440\u0433\u0435\u043d\u0446\u0438\u044e \u043c\u0435\u0436\u0434\u0443 \u0430\u043f\u0440\u0438\u043e\u0440\u043d\u044b\u043c \u0438 \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440\u043d\u044b\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f\u043c\u0438 \u043b\u0430\u0442\u0435\u043d\u0442\u043e\u0432. \u0412\u0441\u0435\u0433\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"KL \\geq 0\" alt=\"KL \\geq 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ac\/ace\/ace3c32727e80abe562d49fe2f95eebc.svg\" width=\"62\" height=\"18\" data-width=\"7.7\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.7ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ac\/ace\/ace3c32727e80abe562d49fe2f95eebc.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/ac\/ace\/ace3c32727e80abe562d49fe2f95eebc.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u043f\u0440\u0438\u0447\u0435\u043c \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u0434\u043e\u0441\u0442\u0438\u0433\u0430\u0435\u0442\u0441\u044f \u043f\u0440\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"q(z) = p_{\\theta}\\left( z\\ |\\ x \\right)\" alt=\"q(z) = p_{\\theta}\\left( z\\ |\\ x \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/caa\/caa474a571019c880a154094c2b67e88.svg\" width=\"122\" height=\"18\" data-width=\"15.19\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 15.19ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/caa\/caa474a571019c880a154094c2b67e88.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ca\/caa\/caa474a571019c880a154094c2b67e88.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0411\u043e\u043b\u0435\u0435 \u0442\u043e\u0433\u043e, \u043c\u044b \u043d\u0435 \u043c\u043e\u0436\u0435\u043c \u043d\u0430\u043f\u0440\u044f\u043c\u0443\u044e \u0441\u0447\u0438\u0442\u0430\u0442\u044c \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"\\ln{p_{\\theta}(x)} = \\ln{\\int_{}^{}{p_{\\theta}\\left( x\\ |\\ z \\right)q(z)dz}}\" alt=\"\\ln{p_{\\theta}(x)} = \\ln{\\int_{}^{}{p_{\\theta}\\left( x\\ |\\ z \\right)q(z)dz}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/65\/655\/65512429bb59fc8663b6b4129ae218ac.svg\" width=\"239\" height=\"41\" data-width=\"29.934\" data-height=\"5.107\" data-vertical-align=\"-1.988\" style=\"width: 29.934ex; height: 5.107ex; vertical-align: -1.988ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/65\/655\/65512429bb59fc8663b6b4129ae218ac.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/65\/655\/65512429bb59fc8663b6b4129ae218ac.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0435\u0441\u043b\u0438 \u043e\u0446\u0435\u043d\u0438\u0432\u0430\u0442\u044c \u043f\u043e \u041c\u043e\u043d\u0442\u0435-\u041a\u0430\u0440\u043b\u043e, \u043e\u0446\u0435\u043d\u043a\u0430 \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u0441\u044f \u0441\u043c\u0435\u0449\u0435\u043d\u043d\u043e\u0439, \u0442.\u043a. \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0441\u043d\u0430\u0440\u0443\u0436\u0438 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0430). \u0410 \u0441\u0447\u0438\u0442\u0430\u0442\u044c \u0441\u043e\u0432\u043c\u0435\u0441\u0442\u043d\u043e\u0435 ELBO &#8212; \u043c\u043e\u0436\u0435\u043c (\u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0437\u0430\u043d\u0435\u0441\u043b\u0438 \u0432\u043d\u0443\u0442\u0440\u044c \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0430, <img decoding=\"async\" class=\"formula inline\" source=\"q\" alt=\"q\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg\" width=\"12\" height=\"12\" data-width=\"1.041\" data-height=\"1.439\" data-vertical-align=\"-0.439\" style=\"width: 1.041ex; height: 1.439ex; vertical-align: -0.439ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/76\/769\/7694f4a66316e53c8cdd9d9954bd611d.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e\u0434\u0431\u0438\u0440\u0430\u0435\u043c \u0441\u0430\u043c\u0438). \u0421\u043d\u043e\u0432\u0430 \u043f\u043e\u043b\u044c\u0437\u0430 \u043e\u0442 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430: \u043f\u0435\u0440\u0435\u0448\u043b\u0438 \u0432 \u0442\u0430\u043a\u0438\u0435 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b, \u0432 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0437\u0430\u0434\u0430\u0447\u0430 \u0440\u0435\u0448\u0430\u0435\u0442\u0441\u044f \u043f\u0440\u043e\u0449\u0435.<\/p>\n<p>\u0412\u043e \u0432\u0441\u0435\u0445 \u0443\u0447\u0435\u0431\u043d\u044b\u0445 \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u0430\u0445, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u044f \u0432\u0438\u0434\u0435\u043b, ELBO \u0432\u044b\u0432\u043e\u0434\u044f\u0442 \u043c\u0435\u0442\u043e\u0434\u043e\u043c \u00ab\u0445\u0438\u0442\u0440\u044b\u0445 \u0435\u0434\u0438\u043d\u0438\u0446\u00bb. \u042f\u0441\u043d\u043e, \u0447\u0442\u043e \u0442\u0430\u043a \u0434\u0435\u043b\u0430\u044e\u0442, \u0447\u0442\u043e\u0431\u044b \u043d\u0435 \u0443\u0433\u043b\u0443\u0431\u043b\u044f\u0442\u044c\u0441\u044f \u0432 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u0438 \u043d\u0435 \u043e\u0442\u0432\u043b\u0435\u043a\u0430\u0442\u044c\u0441\u044f \u043e\u0442 \u0445\u043e\u0434\u0430 \u043a\u0443\u0440\u0441\u0430 \u0440\u0430\u0434\u0438 \u043e\u0434\u043d\u043e\u0433\u043e \u0432\u044b\u0432\u043e\u0434\u0430. \u041d\u043e \u0442\u0430\u043a\u043e\u0435 \u0434\u043e\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u00ab\u0445\u0438\u0442\u0440\u044b\u0435 \u0435\u0434\u0438\u043d\u0438\u0446\u044b\u00bb \u0441\u043a\u0440\u044b\u0432\u0430\u0435\u0442 \u0431\u043e\u043b\u0435\u0435 \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u0443\u044e \u0438\u0434\u0435\u044e \u0438 \u043a\u0440\u0430\u0439\u043d\u0435 \u043d\u0435\u0438\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\ln{p_{\\theta}(x)} = \\left( \\int_{}^{}{q(z)dz} \\right) \\cdot \\ln\\left( \\frac{p_{\\theta}(x,z)}{p_{\\theta}\\left( z\\ |\\ x \\right)} \\cdot \\frac{q(z)}{q(z)} \\right) = \\int_{}^{}{q(z)}\\ln\\frac{p_{\\theta}(x,z)}{q(z)}dz + \\int_{}^{}{q(z)\\ln\\frac{q(z)}{p_{\\theta}\\left( z\\ |\\ x \\right)}dz} = ELBO\\lbrack q\\rbrack + KL(q(z)\\ ||\\ p_{\\theta}\\left( z\\ |\\ x \\right))\" alt=\"\\ln{p_{\\theta}(x)} = \\left( \\int_{}^{}{q(z)dz} \\right) \\cdot \\ln\\left( \\frac{p_{\\theta}(x,z)}{p_{\\theta}\\left( z\\ |\\ x \\right)} \\cdot \\frac{q(z)}{q(z)} \\right) = \\int_{}^{}{q(z)}\\ln\\frac{p_{\\theta}(x,z)}{q(z)}dz + \\int_{}^{}{q(z)\\ln\\frac{q(z)}{p_{\\theta}\\left( z\\ |\\ x \\right)}dz} = ELBO\\lbrack q\\rbrack + KL(q(z)\\ ||\\ p_{\\theta}\\left( z\\ |\\ x \\right))\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bf\/bfe\/bfef837b6bc2786b983be1bc0903f29e.svg\" width=\"1015\" height=\"44\" data-width=\"126.818\" data-height=\"5.475\" data-vertical-align=\"-2.172\" style=\"width: 126.818ex; height: 5.475ex; vertical-align: -2.172ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bf\/bfe\/bfef837b6bc2786b983be1bc0903f29e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/bf\/bfe\/bfef837b6bc2786b983be1bc0903f29e.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u043e\u0441\u0442\u0430\u0435\u0442\u0441\u044f \u0432\u043e\u043f\u0440\u043e\u0441: \u0441\u0447\u0438\u0442\u0430\u0442\u044c \u0438 \u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c ELBO \u0443\u0434\u043e\u0431\u043d\u0435\u0435 \u0442\u0435\u043e\u0440\u0435\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u2013 \u043d\u043e \u043a\u0430\u043a?<\/p>\n<p>\u041f\u0440\u0438\u043c\u0435\u0440 1: \u043a\u043b\u0430\u0441\u0442\u0435\u0440\u0438\u0437\u0430\u0446\u0438\u044f. \u041f\u0443\u0441\u0442\u044c \u0443 \u043d\u0430\u0441 \u0435\u0441\u0442\u044c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0442\u043e\u0447\u0435\u043a, \u0438 \u043d\u0430\u0448\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u2013 \u0440\u0430\u0437\u0434\u0435\u043b\u0438\u0442\u044c \u0438\u0445 \u043d\u0430 \u0433\u0440\u0443\u043f\u043f\u044b \u00ab\u043f\u043e\u0445\u043e\u0436\u0438\u0445\u00bb. \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u0442\u043e\u0447\u043a\u0438 \u043f\u043e\u0440\u043e\u0436\u0434\u0435\u043d\u044b \u0438\u0437 \u0441\u043c\u0435\u0441\u0438 \u0433\u0430\u0443\u0441\u0441\u0438\u0430\u043d, \u0432\u0437\u0432\u0435\u0448\u0435\u043d\u043d\u044b\u0445 \u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u0430\u043b\u044c\u043d\u044b\u043c \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c: <img decoding=\"async\" class=\"formula inline\" source=\"p_{\\theta}(x) = \\sum_{k = 1}^{K}{\\pi_{k}N\\left( x\\ |\\ \\mu_{k},\\Sigma_{k} \\right)}\" alt=\"p_{\\theta}(x) = \\sum_{k = 1}^{K}{\\pi_{k}N\\left( x\\ |\\ \\mu_{k},\\Sigma_{k} \\right)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b4\/6b494f1db6041edc6c3902b16cb02325.svg\" width=\"219\" height=\"54\" data-width=\"27.349\" data-height=\"6.784\" data-vertical-align=\"-2.827\" style=\"width: 27.349ex; height: 6.784ex; vertical-align: -2.827ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b4\/6b494f1db6041edc6c3902b16cb02325.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6b\/6b4\/6b494f1db6041edc6c3902b16cb02325.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041a\u0430\u0436\u0434\u044b\u0439 \u043a\u043b\u0430\u0441\u0442\u0435\u0440 \u043f\u043e\u0440\u043e\u0436\u0434\u0435\u043d \u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u043e\u0439 \u0433\u0430\u0443\u0441\u0441\u0438\u0430\u043d\u043e\u0439, \u043d\u043e \u043c\u044b \u043d\u0435 \u0437\u043d\u0430\u0435\u043c, \u043a\u0430\u043a\u043e\u0439 \u0438\u043c\u0435\u043d\u043d\u043e \u2013 \u044d\u0442\u043e \u043d\u0430\u0448 \u043b\u0430\u0442\u0435\u043d\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"z\\ \\sim\\ Cat(\\pi)\" alt=\"z\\ \\sim\\ Cat(\\pi)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ce\/cec\/cec27290e02378a1cbc5e6e2e9397133.svg\" width=\"96\" height=\"18\" data-width=\"11.983\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 11.983ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ce\/cec\/cec27290e02378a1cbc5e6e2e9397133.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/ce\/cec\/cec27290e02378a1cbc5e6e2e9397133.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412\u044b\u0442\u044f\u043d\u0443\u0432 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043b\u0430\u0442\u0435\u043d\u0442, \u043c\u043e\u0434\u0435\u043b\u044c \u0441\u043e\u043f\u043e\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0435\u043c\u0443 \u043d\u0443\u0436\u043d\u0443\u044e \u0433\u0430\u0443\u0441\u0441\u0438\u0430\u043d\u0443 \u0438 \u0432\u044b\u0442\u044f\u0433\u0438\u0432\u0430\u0435\u0442 \u0442\u043e\u0447\u043a\u0443 \u0438\u0437 \u043d\u0435\u0435: <img decoding=\"async\" class=\"formula inline\" source=\"x\\ \\sim\\ N\\left( \\mu_{z},\\Sigma_{z} \\right)\" alt=\"x\\ \\sim\\ N\\left( \\mu_{z},\\Sigma_{z} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05f\/05fa8ac9381b37adec3c8b0cdd3aba8d.svg\" width=\"124\" height=\"18\" data-width=\"15.456\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 15.456ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05f\/05fa8ac9381b37adec3c8b0cdd3aba8d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05f\/05fa8ac9381b37adec3c8b0cdd3aba8d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>.<\/p>\n<p>\u0418\u0434\u0435\u044f: \u0434\u0430\u0432\u0430\u0439\u0442\u0435 \u0437\u0430\u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u043c \u0442\u0435\u043a\u0443\u0449\u0438\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043c\u043e\u0434\u0435\u043b\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta_{t} = \\left\\{ \\pi,\\mu,\\Sigma \\right\\}_{t}\" alt=\"\\theta_{t} = \\left\\{ \\pi,\\mu,\\Sigma \\right\\}_{t}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbe\/cbe725d9736a34ed7310af7fed988588.svg\" width=\"113\" height=\"19\" data-width=\"14.171\" data-height=\"2.36\" data-vertical-align=\"-0.614\" style=\"width: 14.171ex; height: 2.36ex; vertical-align: -0.614ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbe\/cbe725d9736a34ed7310af7fed988588.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbe\/cbe725d9736a34ed7310af7fed988588.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u0440\u0438 \u043d\u0438\u0445 \u043e\u0431\u043d\u043e\u0432\u0438\u043c \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440: \u0441 \u043a\u0430\u043a\u043e\u0439 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c\u044e \u0442\u043e\u0447\u043a\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"x_{i}\" alt=\"x_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05e\/05e42209d67fe1eb15a055e9d3b3770e.svg\" width=\"16\" height=\"12\" data-width=\"2.034\" data-height=\"1.357\" data-vertical-align=\"-0.357\" style=\"width: 2.034ex; height: 1.357ex; vertical-align: -0.357ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05e\/05e42209d67fe1eb15a055e9d3b3770e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05e\/05e42209d67fe1eb15a055e9d3b3770e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e\u0440\u043e\u0436\u0434\u0435\u043d\u0430 \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"k\" alt=\"k\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg\" width=\"12\" height=\"13\" data-width=\"1.179\" data-height=\"1.595\" data-vertical-align=\"-0.025\" style=\"width: 1.179ex; height: 1.595ex; vertical-align: -0.025ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/8c\/8ce\/8ce4b16b22b58894aa86c421e8759df3.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u043c\u044f\u0433\u043a\u0438\u0435 \u043f\u0440\u0438\u043d\u0430\u0434\u043b\u0435\u0436\u043d\u043e\u0441\u0442\u0438 \u0442\u043e\u0447\u0435\u043a \u043a\u043b\u0430\u0441\u0442\u0435\u0440\u0430\u043c):<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"r_{ik} = p_{\\theta_{t}}\\left( z_{i} = k\\ |\\ x_{i} \\right) = \\frac{\\pi_{k}N\\left( x_{i}\\ |\\ \\mu_{k},\\Sigma_{k} \\right)}{\\sum_{j = 1}^{K}{\\pi_{j}N\\left( x_{i}\\ |\\ \\mu_{j},\\Sigma_{j} \\right)}} = \\frac{\\pi_{k}\\exp\\left( - \\left( x_{i} - \\mu_{k} \\right)^{2}\/2\\sigma^{2} \\right)}{\\sum_{j}^{}{\\pi_{j}\\exp\\left( - \\left( x_{i} - \\mu_{j} \\right)^{2}\/2\\sigma^{2} \\right)}}\" alt=\"r_{ik} = p_{\\theta_{t}}\\left( z_{i} = k\\ |\\ x_{i} \\right) = \\frac{\\pi_{k}N\\left( x_{i}\\ |\\ \\mu_{k},\\Sigma_{k} \\right)}{\\sum_{j = 1}^{K}{\\pi_{j}N\\left( x_{i}\\ |\\ \\mu_{j},\\Sigma_{j} \\right)}} = \\frac{\\pi_{k}\\exp\\left( - \\left( x_{i} - \\mu_{k} \\right)^{2}\/2\\sigma^{2} \\right)}{\\sum_{j}^{}{\\pi_{j}\\exp\\left( - \\left( x_{i} - \\mu_{j} \\right)^{2}\/2\\sigma^{2} \\right)}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f7\/f7b\/f7bd9015e3ca5795a3f9919886dc4910.svg\" width=\"627\" height=\"73\" data-width=\"78.327\" data-height=\"9.09\" data-vertical-align=\"-3.98\" style=\"width: 78.327ex; height: 9.09ex; vertical-align: -3.98ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f7\/f7b\/f7bd9015e3ca5795a3f9919886dc4910.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f7\/f7b\/f7bd9015e3ca5795a3f9919886dc4910.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041e\u0431\u0440\u0430\u0442\u0438\u043c \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435: <img decoding=\"async\" class=\"formula inline\" source=\"r_{ik}\" alt=\"r_{ik}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e68f0d9ab562070faa43f1160adb21.svg\" width=\"21\" height=\"12\" data-width=\"2.594\" data-height=\"1.357\" data-vertical-align=\"-0.357\" style=\"width: 2.594ex; height: 1.357ex; vertical-align: -0.357ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e68f0d9ab562070faa43f1160adb21.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e68f0d9ab562070faa43f1160adb21.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u044d\u0442\u043e softmax \u0441 \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u043e\u0439 <img decoding=\"async\" class=\"formula inline\" source=\"2\\sigma^{2}\" alt=\"2\\sigma^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e60\/e6034e7acc604d113eabac54f6e45160.svg\" width=\"27\" height=\"21\" data-width=\"3.411\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 3.411ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e60\/e6034e7acc604d113eabac54f6e45160.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e60\/e6034e7acc604d113eabac54f6e45160.svg 781w\" loading=\"lazy\" decode=\"async\"\/> (\u0438 \u0432\u0435\u0441\u0430\u043c\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\pi\" alt=\"\\pi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f0\/4f08e3dba63dc6d40b22952c7a9dac6d.svg\" width=\"12\" height=\"18\" data-width=\"1.29\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 1.29ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f0\/4f08e3dba63dc6d40b22952c7a9dac6d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4f\/4f0\/4f08e3dba63dc6d40b22952c7a9dac6d.svg 781w\" loading=\"lazy\" decode=\"async\"\/>). \u0423\u0441\u0442\u0440\u0435\u043c\u0438\u0432 \u0442\u0435\u043c\u043f\u0435\u0440\u0430\u0442\u0443\u0440\u0443 \u043a \u043d\u0443\u043b\u044e, \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u0436\u0435\u0441\u0442\u043a\u0443\u044e \u043f\u0440\u0438\u043d\u0430\u0434\u043b\u0435\u0436\u043d\u043e\u0441\u0442\u044c \u043a\u043b\u0430\u0441\u0442\u0435\u0440\u0430\u043c \u2013 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c k-means.<\/p>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0437\u0430\u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"r_{ik}\" alt=\"r_{ik}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e68f0d9ab562070faa43f1160adb21.svg\" width=\"21\" height=\"12\" data-width=\"2.594\" data-height=\"1.357\" data-vertical-align=\"-0.357\" style=\"width: 2.594ex; height: 1.357ex; vertical-align: -0.357ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e68f0d9ab562070faa43f1160adb21.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e6\/e6e\/e6e68f0d9ab562070faa43f1160adb21.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0438 \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u0435\u043c ELBO:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"ELBO = \\sum_{i,k}^{}{r_{ik}\\ln{p_{\\theta}(x,z)}} - \\sum_{i,k}^{}{r_{ik}\\ln r_{ik}} = \\sum_{i,k}^{}{r_{ik}\\ln\\left( \\pi_{k}N\\left( x_{i}\\ |\\ \\mu_{k},\\Sigma_{k} \\right) \\right)} - \\sum_{i,k}^{}{r_{ik}\\ln r_{ik}}\" alt=\"ELBO = \\sum_{i,k}^{}{r_{ik}\\ln{p_{\\theta}(x,z)}} - \\sum_{i,k}^{}{r_{ik}\\ln r_{ik}} = \\sum_{i,k}^{}{r_{ik}\\ln\\left( \\pi_{k}N\\left( x_{i}\\ |\\ \\mu_{k},\\Sigma_{k} \\right) \\right)} - \\sum_{i,k}^{}{r_{ik}\\ln r_{ik}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6a\/6a5\/6a5bdf43c81fbe9105e646a07c5a5677.svg\" width=\"674\" height=\"47\" data-width=\"84.226\" data-height=\"5.871\" data-vertical-align=\"-2.37\" style=\"width: 84.226ex; height: 5.871ex; vertical-align: -2.37ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6a\/6a5\/6a5bdf43c81fbe9105e646a07c5a5677.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/6a\/6a5\/6a5bdf43c81fbe9105e646a07c5a5677.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u043c ELBO \u043f\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u043c \u043c\u043e\u0434\u0435\u043b\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\theta = \\left\\{ \\pi,\\mu,\\Sigma \\right\\}\" alt=\"\\theta = \\left\\{ \\pi,\\mu,\\Sigma \\right\\}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/11b\/11bab9a7435f27c6858ec62fe310cc28.svg\" width=\"101\" height=\"18\" data-width=\"12.64\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 12.64ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/11b\/11bab9a7435f27c6858ec62fe310cc28.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/11\/11b\/11bab9a7435f27c6858ec62fe310cc28.svg 781w\" loading=\"lazy\" decode=\"async\"\/>:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\pi_{k} = \\frac{\\sum_{i}^{}r_{ik}}{n},\\ \\ \\mu_{k} = \\frac{\\sum_{i}^{}{r_{ik}x_{i}}}{\\sum_{i}^{}r_{ik}},\\ \\ \\Sigma_{k} = \\frac{\\sum_{i}^{}{r_{ik}\\left( x_{i} - \\mu_{k} \\right)\\left( x_{i} - \\mu_{k} \\right)^{T}}}{\\sum_{i}^{}r_{ik}}\" alt=\"\\pi_{k} = \\frac{\\sum_{i}^{}r_{ik}}{n},\\ \\ \\mu_{k} = \\frac{\\sum_{i}^{}{r_{ik}x_{i}}}{\\sum_{i}^{}r_{ik}},\\ \\ \\Sigma_{k} = \\frac{\\sum_{i}^{}{r_{ik}\\left( x_{i} - \\mu_{k} \\right)\\left( x_{i} - \\mu_{k} \\right)^{T}}}{\\sum_{i}^{}r_{ik}}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30c\/30c1808ba85685115ce113ec50b914e0.svg\" width=\"509\" height=\"49\" data-width=\"63.577\" data-height=\"6.136\" data-vertical-align=\"-2.502\" style=\"width: 63.577ex; height: 6.136ex; vertical-align: -2.502ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30c\/30c1808ba85685115ce113ec50b914e0.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30c\/30c1808ba85685115ce113ec50b914e0.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041f\u043e\u0432\u0442\u043e\u0440\u044f\u0435\u043c \u0434\u043e \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438. \u0421\u043f\u0443\u0441\u0442\u044f \u043c\u043d\u043e\u0433\u043e \u0448\u0430\u0433\u043e\u0432, \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u043c \u043c\u043e\u0434\u0435\u043b\u044c, \u0443\u043c\u0435\u044e\u0449\u0443\u044e \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u044c \u0442\u043e\u0447\u043a\u0438 \u043a \u043a\u043b\u0430\u0441\u0442\u0435\u0440\u0430\u043c.<\/p>\n<p>\u0417\u0430\u043c\u0435\u0442\u0438\u043c \u0441\u0432\u044f\u0437\u0438:<\/p>\n<ol>\n<li>\n<p>\u0434\u043b\u044f \u043a\u0430\u0436\u0434\u043e\u0439 \u0442\u043e\u0447\u043a\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"x_{i}\" alt=\"x_{i}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05e\/05e42209d67fe1eb15a055e9d3b3770e.svg\" width=\"16\" height=\"12\" data-width=\"2.034\" data-height=\"1.357\" data-vertical-align=\"-0.357\" style=\"width: 2.034ex; height: 1.357ex; vertical-align: -0.357ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05e\/05e42209d67fe1eb15a055e9d3b3770e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/05\/05e\/05e42209d67fe1eb15a055e9d3b3770e.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u044d\u043d\u0435\u0440\u0433\u0438\u044f \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u044b <img decoding=\"async\" class=\"formula inline\" source=\"\\eta_{i,k} = \\ln\\pi_{k} + \\ln{N\\left( x_{i}\\ |\\ \\mu_{k},\\Sigma_{k} \\right)}\" alt=\"\\eta_{i,k} = \\ln\\pi_{k} + \\ln{N\\left( x_{i}\\ |\\ \\mu_{k},\\Sigma_{k} \\right)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e7\/e73\/e73b4d2edb7fba7b77bdb00f1c70e13c.svg\" width=\"238\" height=\"19\" data-width=\"29.751\" data-height=\"2.347\" data-vertical-align=\"-0.608\" style=\"width: 29.751ex; height: 2.347ex; vertical-align: -0.608ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e7\/e73\/e73b4d2edb7fba7b77bdb00f1c70e13c.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/e\/e7\/e73\/e73b4d2edb7fba7b77bdb00f1c70e13c.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u0438\u043c\u0435\u0435\u0442 \u0432\u0438\u0434 <img decoding=\"async\" class=\"formula inline\" source=\"\\psi\\left( \\eta_{i} \\right) = logsumexp\" alt=\"\\psi\\left( \\eta_{i} \\right) = logsumexp\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/89\/891\/8915a3c41d609937d0dffa536ab5836e.svg\" width=\"153\" height=\"18\" data-width=\"19.17\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 19.17ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/89\/891\/8915a3c41d609937d0dffa536ab5836e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/8\/89\/891\/8915a3c41d609937d0dffa536ab5836e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0430 \u043f\u0440\u0438\u043d\u0430\u0434\u043b\u0435\u0436\u043d\u043e\u0441\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"r_{ik} = \\left\\lbrack \\nabla\\psi\\left( \\eta_{i} \\right) \\right\\rbrack_{k}\" alt=\"r_{ik} = \\left\\lbrack \\nabla\\psi\\left( \\eta_{i} \\right) \\right\\rbrack_{k}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba7\/ba76ca3d1518fdd197bb7874912475a1.svg\" width=\"122\" height=\"19\" data-width=\"15.249\" data-height=\"2.36\" data-vertical-align=\"-0.614\" style=\"width: 15.249ex; height: 2.36ex; vertical-align: -0.614ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba7\/ba76ca3d1518fdd197bb7874912475a1.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/ba\/ba7\/ba76ca3d1518fdd197bb7874912475a1.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p>\u043f\u0435\u0440\u0432\u044b\u0439 \u0448\u0430\u0433 \u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u0438 \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u0438\u0442 \u043c\u043e\u043c\u0435\u043d\u0442\u044b\/\u0441\u0440\u0435\u0434\u043d\u0438\u0435 \u0441\u0440\u0430\u0437\u0443 \u0432 \u043e\u043f\u0442\u0438\u043c\u0443\u043c <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla\\psi\\left( \\eta_{i} \\right) = p_{\\theta_{t}}\\left( z_{i} = k\\ |\\ x_{i} \\right)\" alt=\"\\nabla\\psi\\left( \\eta_{i} \\right) = p_{\\theta_{t}}\\left( z_{i} = k\\ |\\ x_{i} \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/65\/65a\/65af295db1f73a7de2637d750a6c434b.svg\" width=\"199\" height=\"18\" data-width=\"24.912\" data-height=\"2.289\" data-vertical-align=\"-0.579\" style=\"width: 24.912ex; height: 2.289ex; vertical-align: -0.579ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/65\/65a\/65af295db1f73a7de2637d750a6c434b.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/6\/65\/65a\/65af295db1f73a7de2637d750a6c434b.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u042d\u0442\u043e \u0448\u0430\u0433 \u0437\u0435\u0440\u043a\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0441\u043f\u0443\u0441\u043a\u0430 \u043f\u043e\u0434\u044a\u0435\u043c\u0430 \u0434\u043e \u0443\u043f\u043e\u0440\u0430<\/p>\n<\/li>\n<li>\n<p>\u0432\u0442\u043e\u0440\u043e\u0439 \u0448\u0430\u0433 \u043f\u0440\u0438\u0440\u0430\u0432\u043d\u0438\u0432\u0430\u0435\u0442 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u044b\u0435 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0438 \u043c\u043e\u0434\u0435\u043b\u0438 \u043a\u043e \u0432\u0437\u0432\u0435\u0448\u0435\u043d\u043d\u044b\u043c \u044d\u043c\u043f\u0438\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u043c \u0438 \u0434\u0432\u0438\u0433\u0430\u0435\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0435 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b. \u042d\u0442\u043e \u0448\u0430\u0433 \u0441 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u043c \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u043e\u043c, <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla^{2}\\psi^{*} = \u0424\u0438\u0448\u0435\u0440^{- 1}\" alt=\"\\nabla^{2}\\psi^{*} = \u0424\u0438\u0448\u0435\u0440^{- 1}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b3\/b38\/b3887dfd0250ff9cfda2e9e402ae8f49.svg\" width=\"139\" height=\"21\" data-width=\"17.37\" data-height=\"2.565\" data-vertical-align=\"-0.717\" style=\"width: 17.37ex; height: 2.565ex; vertical-align: -0.717ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b3\/b38\/b3887dfd0250ff9cfda2e9e402ae8f49.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/b\/b3\/b38\/b3887dfd0250ff9cfda2e9e402ae8f49.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p>\u043f\u043e\u043b\u043d\u044b\u0439 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u2013 \u044d\u0442\u043e \u043f\u043e\u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u043d\u044b\u0439 \u043f\u043e\u0434\u044a\u0435\u043c ELBO: \u043f\u043e \u043e\u0434\u043d\u0438\u043c \u0435\u0433\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u043c + \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e \u043f\u043e \u0434\u0440\u0443\u0433\u0438\u043c<\/p>\n<\/li>\n<\/ol>\n<p>\u042d\u0442\u043e \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f EM-\u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c:<\/p>\n<ol>\n<li>\n<p>\u0415-\u0448\u0430\u0433: \u0437\u0430\u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u043c \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u044b \u043c\u043e\u0434\u0435\u043b\u0438, \u043f\u0440\u0438 \u043d\u0438\u0445 \u043e\u0431\u043d\u043e\u0432\u0438\u043c \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440 <img decoding=\"async\" class=\"formula inline\" source=\"p_{\\theta_{t}}\\left( z\\ |\\ x \\right)\" alt=\"p_{\\theta_{t}}\\left( z\\ |\\ x \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/321\/321e4dd1a5f48c280d35988ee763063e.svg\" width=\"71\" height=\"18\" data-width=\"8.861\" data-height=\"2.289\" data-vertical-align=\"-0.579\" style=\"width: 8.861ex; height: 2.289ex; vertical-align: -0.579ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/321\/321e4dd1a5f48c280d35988ee763063e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/32\/321\/321e4dd1a5f48c280d35988ee763063e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043c\u044b \u0432\u044b\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c <img decoding=\"async\" class=\"formula inline\" source=\"q = p_{\\theta_{t}}\\left( z\\ |\\ x \\right)\" alt=\"q = p_{\\theta_{t}}\\left( z\\ |\\ x \\right)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/700\/700f94b9ca9da09371f3d6574cb823b8.svg\" width=\"103\" height=\"18\" data-width=\"12.919\" data-height=\"2.289\" data-vertical-align=\"-0.579\" style=\"width: 12.919ex; height: 2.289ex; vertical-align: -0.579ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/700\/700f94b9ca9da09371f3d6574cb823b8.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/700\/700f94b9ca9da09371f3d6574cb823b8.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442\u043e \u0437\u0430\u0437\u043e\u0440 <img decoding=\"async\" class=\"formula inline\" source=\"KL\\left( q(z)\\ ||\\ p_{\\theta}\\left( z\\ |\\ x \\right) \\right) = 0\" alt=\"KL\\left( q(z)\\ ||\\ p_{\\theta}\\left( z\\ |\\ x \\right) \\right) = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/21e\/21e845931fc6f5862f92f8cac3167a00.svg\" width=\"195\" height=\"18\" data-width=\"24.399\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 24.399ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/21e\/21e845931fc6f5862f92f8cac3167a00.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/21\/21e\/21e845931fc6f5862f92f8cac3167a00.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0433\u0440\u0430\u043d\u0438\u0446\u0430 ELBO \u043a\u0430\u0441\u0430\u0435\u0442\u0441\u044f <img decoding=\"async\" class=\"formula inline\" source=\"\\ln{p_{\\theta}(x)}\" alt=\"\\ln{p_{\\theta}(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a49265f4d9238fd33fd0ea32b1a04e6d.svg\" width=\"59\" height=\"18\" data-width=\"7.394\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 7.394ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a49265f4d9238fd33fd0ea32b1a04e6d.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/a\/a4\/a49\/a49265f4d9238fd33fd0ea32b1a04e6d.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<\/li>\n<li>\n<p>\u041c-\u0448\u0430\u0433: \u0444\u0438\u043a\u0441\u0438\u0440\u0443\u0435\u043c \u043b\u0430\u0442\u0435\u043d\u0442, \u0441\u0447\u0438\u0442\u0430\u0435\u043c ELBO \u0438 \u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0438\u0440\u0443\u0435\u043c \u043f\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u043c. \u041e\u0442\u0441\u044e\u0434\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"\\ln{p_{\\theta_{t}}(x)} \\leq \\ln{p_{\\theta_{t + 1}}(x)}\" alt=\"\\ln{p_{\\theta_{t}}(x)} \\leq \\ln{p_{\\theta_{t + 1}}(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbf\/cbfb936edb7a4a758015bbd74256fc96.svg\" width=\"163\" height=\"19\" data-width=\"20.334\" data-height=\"2.369\" data-vertical-align=\"-0.619\" style=\"width: 20.334ex; height: 2.369ex; vertical-align: -0.619ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbf\/cbfb936edb7a4a758015bbd74256fc96.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/cb\/cbf\/cbfb936edb7a4a758015bbd74256fc96.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0442.\u0435. \u043f\u0440\u0430\u0432\u0434\u043e\u043f\u043e\u0434\u043e\u0431\u0438\u0435 \u0440\u0430\u0441\u0442\u0435\u0442 \u043c\u043e\u043d\u043e\u0442\u043e\u043d\u043d\u043e<\/p>\n<\/li>\n<\/ol>\n<p>\u041f\u0440\u0438\u043c\u0435\u0440 2: \u0433\u0435\u043d\u0435\u0440\u0430\u0446\u0438\u044f \u043a\u0430\u0440\u0442\u0438\u043d\u043e\u043a. \u041f\u0443\u0441\u0442\u044c \u0443 \u043d\u0430\u0441 \u0435\u0441\u0442\u044c \u0434\u0430\u0442\u0430\u0441\u0435\u0442 \u0438\u0437 \u043a\u0430\u0440\u0442\u0438\u043d\u043e\u043a, \u0437\u0430\u0434\u0430\u0447\u0430 &#8212; \u043d\u0430\u0443\u0447\u0438\u0442\u044c\u0441\u044f \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043f\u043e\u0445\u043e\u0436\u0438\u0435<\/p>\n<p>\u0418\u0434\u0435\u044f: \u0432\u043e\u0437\u044c\u043c\u0435\u043c \u0430\u0432\u0442\u043e\u044d\u043d\u043a\u043e\u0434\u0435\u0440 &#8212; \u043d\u0435\u0439\u0440\u043e\u0441\u0435\u0442\u044c \u0438\u0437 \u0434\u0432\u0443\u0445 \u0447\u0430\u0441\u0442\u0435\u0439, \u044d\u043d\u043a\u043e\u0434\u0435\u0440\u0430 \u0438 \u0434\u0435\u043a\u043e\u0434\u0435\u0440\u0430. \u0421\u043d\u0430\u0447\u0430\u043b\u0430 \u044d\u043d\u043a\u043e\u0434\u0435\u0440 \u043e\u0442\u043e\u0431\u0440\u0430\u0437\u0438\u0442 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443 \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e \u043c\u0430\u043b\u043e\u0439 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438, \u0430 \u043f\u043e\u0442\u043e\u043c \u0434\u0435\u043a\u043e\u0434\u0435\u0440 \u043f\u043e\u043f\u0440\u043e\u0431\u0443\u0435\u0442 \u0435\u0435 \u043e\u0442\u0442\u0443\u0434\u0430 \u0432\u043e\u0441\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u044c:<\/p>\n<figure class=\"\"><img decoding=\"async\" src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/7c1\/163\/eee\/7c1163eee791cc1eaaff018738872465.png\" width=\"448\" height=\"316\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/7c1\/163\/eee\/7c1163eee791cc1eaaff018738872465.png 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/7c1\/163\/eee\/7c1163eee791cc1eaaff018738872465.png 781w\" loading=\"lazy\" decode=\"async\"\/><\/figure>\n<p>\u041e\u0431\u0443\u0447\u0430\u0442\u044c \u0431\u0443\u0434\u0435\u043c \u0441\u043e \u0448\u0442\u0440\u0430\u0444\u043e\u043c \u0437\u0430 \u043f\u043e\u043f\u0438\u043a\u0441\u0435\u043b\u044c\u043d\u043e\u0435 \u043e\u0442\u043b\u0438\u0447\u0438\u0435 \u043a\u0430\u0440\u0442\u0438\u043d\u043e\u043a. \u0410 \u043f\u043e\u0442\u043e\u043c \u0432\u043e\u0437\u044c\u043c\u0435\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u044b\u0439 \u0432\u0435\u043a\u0442\u043e\u0440 z, \u043f\u0440\u043e\u0433\u043e\u043d\u0438\u043c \u0435\u0433\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0447\u0435\u0440\u0435\u0437 \u0434\u0435\u043a\u043e\u0434\u0435\u0440, \u0438 \u043d\u0430 \u0432\u044b\u0445\u043e\u0434\u0435 \u0443\u0432\u0438\u0434\u0438\u043c \u043a\u0430\u043a\u0443\u044e-\u0442\u043e \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443. \u041e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f, \u0442\u0430\u043a\u043e\u0439 \u043f\u043e\u0434\u0445\u043e\u0434 \u00ab\u0432 \u043b\u043e\u0431\u00bb \u043d\u0435 \u0441\u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043e\u0441\u043c\u044b\u0441\u043b\u0435\u043d\u043d\u044b\u0445 z \u0443\u0441\u0442\u0440\u043e\u0435\u043d\u043e \u0441\u043b\u043e\u0436\u043d\u043e. \u0411\u0435\u0440\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u044b\u0435 z, \u043c\u044b \u0432\u043e\u043e\u0431\u0449\u0435 \u043c\u043e\u0435\u043c \u043d\u0435 \u043f\u043e\u043f\u0430\u0441\u0442\u044c \u0432 \u043e\u0431\u043b\u0430\u0441\u0442\u044c, \u043e\u0442\u043a\u0443\u0434\u0430 \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u0445\u043e\u0440\u043e\u0448\u0438\u0435 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0438.<\/p>\n<p>\u041f\u043e\u043f\u0440\u0430\u0432\u043a\u0430 \u043a \u0438\u0434\u0435\u0435: \u0434\u043e\u0431\u0430\u0432\u0438\u043c \u0432 \u043b\u043e\u0441\u0441 \u0448\u0442\u0440\u0430\u0444, \u0447\u0442\u043e\u0431\u044b z \u0431\u044b\u043b\u0438 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u044b \u043a\u0430\u043a-\u043d\u0438\u0431\u0443\u0434\u044c \u0443\u0434\u043e\u0431\u043d\u043e, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e, \u0430 \u043f\u0440\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0438 \u0431\u0443\u0434\u0435\u043c \u0441\u0435\u043c\u043f\u043b\u0438\u0440\u043e\u0432\u0430\u0442\u044c z \u0438\u0437 \u044d\u0442\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0438 \u0443\u0442\u043e\u0447\u043d\u044f\u0442\u044c \u0435\u0433\u043e \u0444\u043e\u0440\u043c\u0443:<\/p>\n<figure class=\"\"><img decoding=\"async\" src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/a29\/56e\/e2c\/a2956ee2c89bdee0c965c4003bfd8281.png\" width=\"493\" height=\"360\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/a29\/56e\/e2c\/a2956ee2c89bdee0c965c4003bfd8281.png 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/a29\/56e\/e2c\/a2956ee2c89bdee0c965c4003bfd8281.png 781w\" loading=\"lazy\" decode=\"async\"\/><\/figure>\n<p>\u0410\u043d\u0430\u043b\u043e\u0433 \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f\u043c\u0438 \u2013 \u044d\u0442\u043e KL-\u0434\u0438\u0432\u0435\u0440\u0433\u0435\u043d\u0446\u0438\u044f. \u0412 \u043b\u043e\u0441\u0441\u0435 \u0434\u0432\u0430 \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0445: \u043f\u0435\u0440\u0432\u043e\u0435 \u2013 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u043e \u0440\u0435\u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u0438, \u0432\u0442\u043e\u0440\u043e\u0435 \u2013 \u0440\u0435\u0433\u0443\u043b\u044f\u0440\u0438\u0437\u0430\u0446\u0438\u044f \u0437\u0430 \u0444\u043e\u0440\u043c\u0443 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043b\u0430\u0442\u0435\u043d\u0442\u043e\u0432.<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"- Loss = ELBO = \\mathbb{E}_{z\\ \\sim\\ q(z)}\\left\\lbrack \\ln{p_{\\theta}\\left( x\\ |\\ z \\right)} \\right\\rbrack - KL(q\\left( z\\ |\\ x \\right)\\ ||\\ N(0,I))\" alt=\"- Loss = ELBO = \\mathbb{E}_{z\\ \\sim\\ q(z)}\\left\\lbrack \\ln{p_{\\theta}\\left( x\\ |\\ z \\right)} \\right\\rbrack - KL(q\\left( z\\ |\\ x \\right)\\ ||\\ N(0,I))\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/70e\/70e0dbf110feee1e30196f7f6e57f9c6.svg\" width=\"508\" height=\"20\" data-width=\"63.494\" data-height=\"2.497\" data-vertical-align=\"-0.683\" style=\"width: 63.494ex; height: 2.497ex; vertical-align: -0.683ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/70e\/70e0dbf110feee1e30196f7f6e57f9c6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/70\/70e\/70e0dbf110feee1e30196f7f6e57f9c6.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u0422\u0430\u043a\u0430\u044f \u043c\u043e\u0434\u0435\u043b\u044c \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0432\u0430\u0440\u0438\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0439 \u0430\u0432\u0442\u043e\u044d\u043d\u043a\u043e\u0434\u0435\u0440 (VAE) \u2013 \u0438 \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0443\u0436\u0435 \u0437\u0430\u043c\u0435\u0442\u043d\u043e \u043b\u0443\u0447\u0448\u0435. \u0411\u043e\u043b\u0435\u0435 \u0442\u043e\u0433\u043e, \u0430\u0440\u0445\u0438\u0442\u0435\u043a\u0442\u0443\u0440\u0430 VAE \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u043f\u043b\u0430\u0432\u043d\u043e \u043f\u0435\u0440\u0435\u043c\u0435\u0449\u0430\u0442\u044c\u0441\u044f \u043f\u043e \u043b\u0430\u0442\u0435\u043d\u0442\u043d\u043e\u043c\u0443 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0443:<\/p>\n<figure class=\"\"><img decoding=\"async\" src=\"https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/959\/041\/359\/959041359bdab292584d5eb5f820359d.png\" width=\"498\" height=\"508\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/r\/w780\/getpro\/habr\/upload_files\/959\/041\/359\/959041359bdab292584d5eb5f820359d.png 780w,&#10;       https:\/\/habrastorage.org\/r\/w1560\/getpro\/habr\/upload_files\/959\/041\/359\/959041359bdab292584d5eb5f820359d.png 781w\" loading=\"lazy\" decode=\"async\"\/><\/figure>\n<p>\u042d\u043d\u043a\u043e\u0434\u0435\u0440 \u0432\u044b\u0434\u0430\u0435\u0442 \u043c\u043e\u043c\u0435\u043d\u0442\u043d\u0443\u044e \u0441\u0442\u043e\u0440\u043e\u043d\u0443 \u043b\u0430\u0442\u0435\u043d\u0442\u043d\u043e\u0433\u043e \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440\u0430 \u0438 \u0443\u0447\u0438\u0442\u0441\u044f \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0430\u0442\u044c <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla\\psi\" alt=\"\\nabla\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/92a\/92aff5e1c07e11789334c0844b8d9301.svg\" width=\"27\" height=\"18\" data-width=\"3.357\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.357ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/92a\/92aff5e1c07e11789334c0844b8d9301.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/92\/92a\/92aff5e1c07e11789334c0844b8d9301.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u2013 \u043e\u0442\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0438\u0437 \u044d\u043d\u0435\u0440\u0433\u0438\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"\\ln{p_{\\theta}(x,z)}\" alt=\"\\ln{p_{\\theta}(x,z)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/98\/98a\/98a15911bd55dbce864c27656463b734.svg\" width=\"76\" height=\"18\" data-width=\"9.452\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 9.452ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/98\/98a\/98a15911bd55dbce864c27656463b734.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/9\/98\/98a\/98a15911bd55dbce864c27656463b734.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0432 \u0430\u043f\u043e\u0441\u0442\u0435\u0440\u0438\u043e\u0440. \u0414\u0435\u043a\u043e\u0434\u0435\u0440 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438\u0437\u0443\u0435\u0442 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u0443\u044e \u0441\u0442\u043e\u0440\u043e\u043d\u0443 \u0432\u044b\u0445\u043e\u0434\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0438 \u0443\u0447\u0438\u0442\u0441\u044f \u0430\u043f\u043f\u0440\u043e\u043a\u0441\u0438\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u044d\u043d\u0435\u0440\u0433\u0438\u044e. \u0421\u0430\u043c\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0432 ELBO<\/p>\n<p>\u041f\u0440\u0438\u043c\u0435\u0440 3: \u0434\u0438\u0444\u0444\u0443\u0437\u0438\u043e\u043d\u043d\u044b\u0435 \u043c\u043e\u0434\u0435\u043b\u0438. \u042d\u0442\u043e \u043f\u0440\u0438\u043d\u0446\u0438\u043f\u0438\u0430\u043b\u044c\u043d\u043e \u0434\u0440\u0443\u0433\u043e\u0439 \u043f\u043e\u0434\u043a\u043b\u0430\u0441\u0441 \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u044b\u0445 \u043c\u043e\u0434\u0435\u043b\u0435\u0439, \u043e\u043f\u0438\u0440\u0430\u044e\u0449\u0438\u0439\u0441\u044f \u043d\u0430 \u0442\u0435\u043e\u0440\u0438\u044e \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0443\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u0438 \u0441\u0442\u043e\u0445.\u0434\u0438\u0444\u0443\u0440\u044b \u2013 \u0442\u0435\u043c\u044b \u043d\u0430\u043c\u043d\u043e\u0433\u043e \u0431\u043e\u043b\u0435\u0435 \u0441\u043b\u043e\u0436\u043d\u044b\u0435 \u0434\u043b\u044f \u043f\u043e\u0433\u0440\u0443\u0436\u0435\u043d\u0438\u044f. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043e\u0447\u0435\u043d\u044c \u043f\u043e \u0432\u0435\u0440\u0445\u0430\u043c \u0438 \u043b\u0438\u0448\u044c \u043f\u043e\u043a\u0430\u0436\u0435\u043c, \u0433\u0434\u0435 \u0442\u0430\u043c \u041b\u0435\u0436\u0430\u043d\u0434\u0440.<\/p>\n<p>\u0421\u043d\u043e\u0432\u0430 \u0445\u043e\u0442\u0438\u043c \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0438. \u0418\u0434\u0435\u044f:<\/p>\n<ol>\n<li>\n<p>\u0431\u0435\u0440\u0435\u043c \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443 \u0438 \u043f\u043e\u0441\u0442\u0435\u043f\u0435\u043d\u043d\u043e \u0434\u043e\u0431\u0430\u0432\u043b\u044f\u0435\u043c \u043a \u043d\u0435\u0439 \u0431\u0435\u043b\u044b\u0439 \u0448\u0443\u043c, \u043f\u043e\u043a\u0430 \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0430 \u043d\u0435 \u0438\u0441\u0447\u0435\u0437\u043d\u0435\u0442 (\u043f\u0440\u044f\u043c\u043e\u0439 \u043f\u0440\u043e\u0446\u0435\u0441\u0441)<\/p>\n<\/li>\n<li>\n<p>\u043d\u0430\u0443\u0447\u0438\u043c \u043c\u043e\u0434\u0435\u043b\u044c \u0432\u043e\u0441\u0441\u0442\u0430\u043d\u0430\u0432\u043b\u0438\u0432\u0430\u0442\u044c \u0438\u0441\u0445\u043e\u0434\u043d\u0443\u044e \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443 \u0438\u0437 \u0448\u0443\u043c\u0430 (\u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0439 \u043f\u0440\u043e\u0446\u0435\u0441\u0441). \u041e\u0431\u0443\u0447\u0430\u0435\u043c \u0441\u043d\u043e\u0432\u0430 \u0447\u0435\u0440\u0435\u0437 \u043c\u0430\u043a\u0441\u0438\u043c\u0438\u0437\u0430\u0446\u0438\u044e ELBO. \u041e\u0431\u0443\u0447\u0435\u043d\u043d\u043e\u0439 \u043c\u043e\u0434\u0435\u043b\u0438 \u043f\u043e\u0434\u0430\u0435\u043c \u043d\u0430 \u0432\u0445\u043e\u0434 \u0442\u043e\u043b\u044c\u043a\u043e \u0448\u0443\u043c, \u0430 \u043d\u0430 \u0432\u044b\u0445\u043e\u0434\u0435 &#8212; \u0440\u0435\u0430\u043b\u0438\u0441\u0442\u0438\u0447\u043d\u0430\u044f \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0430.<\/p>\n<\/li>\n<\/ol>\n<p>\u0412 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u043c \u043f\u0440\u0435\u0434\u0435\u043b\u0435, \u043f\u0440\u044f\u043c\u043e\u0439 \u043f\u0440\u043e\u0446\u0435\u0441\u0441 \u2013 \u0421\u0414\u0423: <img decoding=\"async\" class=\"formula inline\" source=\"dx = f(x,t)dt + g(t)dw\" alt=\"dx = f(x,t)dt + g(t)dw\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0a\/0a4\/0a4c4446b606b7e2aa01d6a4cabca524.svg\" width=\"183\" height=\"18\" data-width=\"22.821\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 22.821ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0a\/0a4\/0a4c4446b606b7e2aa01d6a4cabca524.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/0\/0a\/0a4\/0a4c4446b606b7e2aa01d6a4cabca524.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041e\u0431\u0440\u0430\u0449\u0435\u043d\u0438\u0435 \u0432\u0440\u0435\u043c\u0435\u043d\u0438 \u2013 \u0441\u043d\u043e\u0432\u0430 \u0434\u0438\u0444\u0444\u0443\u0437\u0438\u044f: <img decoding=\"async\" class=\"formula inline\" source=\"dx = \\left\\lbrack f - g^{2}\\nabla_{x}\\ln{p_{t}(x)} \\right\\rbrack dt + g(t)d\\overline{w},\\ \\ \\ \\frac{dx}{dt} = f - \\frac{1}{2}g^{2}\\nabla_{x}\\ln{p_{t}(x)}\" alt=\"dx = \\left\\lbrack f - g^{2}\\nabla_{x}\\ln{p_{t}(x)} \\right\\rbrack dt + g(t)d\\overline{w},\\ \\ \\ \\frac{dx}{dt} = f - \\frac{1}{2}g^{2}\\nabla_{x}\\ln{p_{t}(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f84\/f845af405cb8aadde225362ebad18344.svg\" width=\"506\" height=\"37\" data-width=\"63.225\" data-height=\"4.676\" data-vertical-align=\"-1.773\" style=\"width: 63.225ex; height: 4.676ex; vertical-align: -1.773ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f84\/f845af405cb8aadde225362ebad18344.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f8\/f84\/f845af405cb8aadde225362ebad18344.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0412\u0441\u0435 \u043d\u0435\u0438\u0437\u0432\u0435\u0441\u0442\u043d\u043e\u0435 \u0437\u0434\u0435\u0441\u044c \u2013 \u0441\u043a\u043e\u0440 <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla_{x}\\ln{p_{t}(x)}\" alt=\"\\nabla_{x}\\ln{p_{t}(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c4\/7c49c3b87f57205fc195e8880a260ca2.svg\" width=\"85\" height=\"18\" data-width=\"10.586\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 10.586ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c4\/7c49c3b87f57205fc195e8880a260ca2.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/7c\/7c4\/7c49c3b87f57205fc195e8880a260ca2.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0435\u0433\u043e \u0438 \u0443\u0447\u0438\u0442 \u043d\u0435\u0439\u0440\u043e\u0441\u0435\u0442\u044c. \u0421\u0433\u0435\u043d\u0435\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443 = \u043f\u0440\u043e\u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0435 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435, \u0434\u0432\u0438\u0433\u0430\u044f\u0441\u044c \u043f\u043e \u0441\u043a\u043e\u0440\u0443. \u041f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043d\u0430 \u0433\u0435\u043d\u0435\u0440\u0430\u0446\u0438\u044e \u043a\u0430\u043a \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0443 \u043e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0443\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f: \u043a \u0448\u0443\u043c\u043e\u0432\u043e\u043c\u0443 \u0421\u0414\u0423 \u0434\u043e\u0431\u0430\u0432\u0438\u043c \u0440\u0443\u043b\u044f\u0449\u0438\u0439 \u0441\u043d\u043e\u0441 <img decoding=\"async\" class=\"formula inline\" source=\"u(x,t)\" alt=\"u(x,t)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30b\/30bfdcfbd9a4e20bba0148762f16c555.svg\" width=\"49\" height=\"18\" data-width=\"6.171\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.171ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30b\/30bfdcfbd9a4e20bba0148762f16c555.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/3\/30\/30b\/30bfdcfbd9a4e20bba0148762f16c555.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0447\u0442\u043e\u0431\u044b \u043a \u043a\u043e\u043d\u0446\u0443 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u044b \u043f\u0440\u0438\u0439\u0442\u0438 \u043d\u0430 \u0434\u0430\u043d\u043d\u044b\u0435, \u0441 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u043e\u0439 \u0446\u0435\u043d\u043e\u0439 \u0440\u0443\u043b\u0435\u043d\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"l(u) = \\frac{1}{2}u^{2}\" alt=\"l(u) = \\frac{1}{2}u^{2}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/24\/240\/240159759d896b4801e4e728092ce048.svg\" width=\"89\" height=\"37\" data-width=\"11.154\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 11.154ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/24\/240\/240159759d896b4801e4e728092ce048.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/2\/24\/240\/240159759d896b4801e4e728092ce048.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0424\u0443\u043d\u043a\u0446\u0438\u044f \u0446\u0435\u043d\u043d\u043e\u0441\u0442\u0438 <img decoding=\"async\" class=\"formula inline\" source=\"V(x,t)\" alt=\"V(x,t)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1a\/1ae\/1ae919ef6ec1d1f5f9b2a5a219aa1601.svg\" width=\"53\" height=\"18\" data-width=\"6.617\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.617ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1a\/1ae\/1ae919ef6ec1d1f5f9b2a5a219aa1601.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/1a\/1ae\/1ae919ef6ec1d1f5f9b2a5a219aa1601.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043f\u043e\u0434\u0447\u0438\u043d\u044f\u0435\u0442\u0441\u044f \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044e \u0413\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0430-\u042f\u043a\u043e\u0431\u0438-\u0411\u0435\u043b\u043c\u0430\u043d\u0430:<\/p>\n<p><img decoding=\"async\" class=\"formula\" source=\"\\partial_{t}V + \\inf_{u}\\left\\lbrack \\frac{1}{2}u^{2} + (f + gu)\\nabla V \\right\\rbrack + \\frac{1}{2}g^{2}\\Delta V = 0\" alt=\"\\partial_{t}V + \\inf_{u}\\left\\lbrack \\frac{1}{2}u^{2} + (f + gu)\\nabla V \\right\\rbrack + \\frac{1}{2}g^{2}\\Delta V = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/156\/15612d57f430f7cc16e49597cffc2f11.svg\" width=\"365\" height=\"43\" data-width=\"45.641\" data-height=\"5.428\" data-vertical-align=\"-2.148\" style=\"width: 45.641ex; height: 5.428ex; vertical-align: -2.148ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/156\/15612d57f430f7cc16e49597cffc2f11.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/15\/156\/15612d57f430f7cc16e49597cffc2f11.svg 781w\" loading=\"lazy\" decode=\"async\"\/><\/p>\n<p>\u041c\u044b \u0443\u0436\u0435 \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u043b\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 \u0413\u042f \u0432 \u0444\u0438\u0437\u0438\u043a\u0435, \u0430 \u0413\u042f\u0411 \u2013 \u0435\u0433\u043e \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u043d\u044b\u0439 \u0430\u043d\u0430\u043b\u043e\u0433. \u0412\u043d\u0443\u0442\u0440\u0435\u043d\u043d\u0438\u0439 \u0438\u043d\u0444\u0438\u043c\u0443\u043c \u2013 \u044d\u0442\u043e \u0431\u0443\u043a\u0432\u0430\u043b\u044c\u043d\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043e\u0442 \u0446\u0435\u043d\u044b \u0440\u0443\u043b\u0435\u043d\u0438\u044f <img decoding=\"async\" class=\"formula inline\" source=\"l(u)\" alt=\"l(u)\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4c\/4cd\/4cd3453161fbe0ddd89c6e17afb28602.svg\" width=\"30\" height=\"18\" data-width=\"3.729\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.729ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4c\/4cd\/4cd3453161fbe0ddd89c6e17afb28602.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/4\/4c\/4cd\/4cd3453161fbe0ddd89c6e17afb28602.svg 781w\" loading=\"lazy\" decode=\"async\"\/>, \u0432\u0437\u044f\u0442\u043e\u0435 \u0432 \u0442\u043e\u0447\u043a\u0435 <img decoding=\"async\" class=\"formula inline\" source=\"- g\\nabla V\" alt=\"- g\\nabla V\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/73c\/73cbad8bff01a35f505e64180daba5a3.svg\" width=\"52\" height=\"18\" data-width=\"6.464\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 6.464ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/73c\/73cbad8bff01a35f505e64180daba5a3.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/73\/73c\/73cbad8bff01a35f505e64180daba5a3.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0420\u0435\u0448\u0430\u0435\u043c, \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u00ab\u0443\u043f\u0440\u0430\u0432\u043b\u044f\u044e\u0449\u0438\u0439 \u0433\u0430\u043c\u0438\u043b\u044c\u0442\u043e\u043d\u0438\u0430\u043d\u00bb: <img decoding=\"async\" class=\"formula inline\" source=\"H(x,p) = fp - \\frac{1}{2}g^{2}p^{2},\\ \\ \\ \\partial_{t}V + H(x,\\nabla V) + \\frac{1}{2}g^{2}\\Delta V = 0\" alt=\"H(x,p) = fp - \\frac{1}{2}g^{2}p^{2},\\ \\ \\ \\partial_{t}V + H(x,\\nabla V) + \\frac{1}{2}g^{2}\\Delta V = 0\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f36\/f366211d1b4f571905c8854c4633e5cd.svg\" width=\"442\" height=\"37\" data-width=\"55.293\" data-height=\"4.588\" data-vertical-align=\"-1.729\" style=\"width: 55.293ex; height: 4.588ex; vertical-align: -1.729ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f36\/f366211d1b4f571905c8854c4633e5cd.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/f\/f3\/f36\/f366211d1b4f571905c8854c4633e5cd.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u0417\u0430\u043c\u0435\u043d\u0430 <img decoding=\"async\" class=\"formula inline\" source=\"V = - \\ln\\psi\" alt=\"V = - \\ln\\psi\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/10e\/10e3fe9c85abaa96e38ad4b6fc1b7ca6.svg\" width=\"85\" height=\"18\" data-width=\"10.631\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 10.631ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/10e\/10e3fe9c85abaa96e38ad4b6fc1b7ca6.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/1\/10\/10e\/10e3fe9c85abaa96e38ad4b6fc1b7ca6.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u043e\u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043b\u044f\u0435\u0442 <img decoding=\"async\" class=\"formula inline\" source=\"\\nabla V\" alt=\"\\nabla V\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c7\/c77\/c7766d55920ce59ded321ce203026e61.svg\" width=\"29\" height=\"18\" data-width=\"3.624\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 3.624ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c7\/c77\/c7766d55920ce59ded321ce203026e61.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/c\/c7\/c77\/c7766d55920ce59ded321ce203026e61.svg 781w\" loading=\"lazy\" decode=\"async\"\/> \u0441\u043e \u0441\u043a\u043e\u0440\u043e\u043c, \u0440\u0435\u0448\u0435\u043d\u0438\u0435: <img decoding=\"async\" class=\"formula inline\" source=\"u^{*} = - g\\nabla V = g\\nabla\\ln\\psi = g\\nabla\\ln{p_{t}(x)}\" alt=\"u^{*} = - g\\nabla V = g\\nabla\\ln\\psi = g\\nabla\\ln{p_{t}(x)}\" src=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/74\/74e\/74eb67a538eeb826bd502c20d60d8a7e.svg\" width=\"283\" height=\"18\" data-width=\"35.437\" data-height=\"2.262\" data-vertical-align=\"-0.566\" style=\"width: 35.437ex; height: 2.262ex; vertical-align: -0.566ex\" sizes=\"auto, (max-width: 780px) 100vw, 50vw\" srcset=\"https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/74\/74e\/74eb67a538eeb826bd502c20d60d8a7e.svg 780w,&#10;       https:\/\/habrastorage.org\/getpro\/habr\/formulas\/7\/74\/74e\/74eb67a538eeb826bd502c20d60d8a7e.svg 781w\" loading=\"lazy\" decode=\"async\"\/>. \u041f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c \u0432 \u0443\u043f\u0440\u0430\u0432\u043b\u044f\u0435\u043c\u043e\u0435 \u0421\u0414\u0423 \u0438 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u0434\u0438\u0444\u0444\u0443\u0437\u0438\u043e\u043d\u043d\u0443\u044e \u043c\u043e\u0434\u0435\u043b\u044c.<\/p>\n<p>\u042f \u043e\u043f\u0443\u0441\u0442\u0438\u043b \u043e\u0447\u0435\u043d\u044c \u043c\u043d\u043e\u0433\u043e \u0434\u0435\u0442\u0430\u043b\u0435\u0439, \u043d\u043e \u0444\u0430\u043a\u0442 \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u041b\u0435\u0436\u0430\u043d\u0434\u0440\u0430 \u043b\u0435\u0436\u0438\u0442 \u0432 \u043e\u0441\u043d\u043e\u0432\u0435 \u043c\u043e\u0434\u0435\u043b\u0438<\/p>\n<p>\u041a\u0430\u043a-\u0442\u043e \u0442\u0430\u043a. \u041e\u043f\u044f\u0442\u044c \u0436\u0435, \u043f\u0440\u0438\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e \u043b\u044e\u0431\u044b\u0435 \u043a\u043e\u043c\u043c\u0435\u043d\u0442\u0430\u0440\u0438\u0438 \u0438 \u0443\u043a\u0430\u0437\u0430\u043d\u0438\u044f \u043d\u0430 \u043e\u0448\u0438\u0431\u043a\u0438\/\u043d\u0435\u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438<\/p>\n<\/div>\n<p>\u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u043e\u0440\u0438\u0433\u0438\u043d\u0430\u043b \u0441\u0442\u0430\u0442\u044c\u0438 <a href=\"https:\/\/habr.com\/ru\/articles\/1080734\/\">https:\/\/habr.com\/ru\/articles\/1080734\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0423\u0447\u0430\u0441\u044c \u0432 \u0438\u043d\u0441\u0442\u0438\u0442\u0443\u0442\u0435, \u044f 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