\u56fd\u5e86\u6ca1\u4eba\u627e\u6211\u73a9\uff0c\u597d\u65e0\u804a\uff0c\u501f\u6b64\u4e5f\u68b3\u7406\u4e00\u4e0b\u5173\u4e8eStein's lemma\u7684\u76f8\u5173\u5185\u5bb9\u3002<\/p>\n
\u9996\u5148\uff0c\u4ecb\u7ecd\u4e00\u4e0b\u8fd9\u4e2a\u5f15\u7406\u3002\u5355\u53d8\u91cf\u7684\u60c5\u51b5\u4e0b\uff0c$X\\sim{\\cal N}\\left( {\\mu ,{\\sigma ^2}} \\right)$\uff0c\u4e14$g\\left( X \\right):{{\\mathbb R}^1} \\to {{\\mathbb R}^1}$\uff0c${\\mathbb E}\\left( {g\\left( X \\right)} \\right)$\u4ee5\u53ca${\\mathbb E}\\left( {g'\\left( X \\right)} \\right)$\u4e3a\u6709\u754c\uff0c\u5219\n$${\\mathbb E}\\left( {g\\left( X \\right)\\left( {X - \\mu } \\right)} \\right) = {\\sigma ^2}{\\mathbb E}\\left( {g'\\left( X \\right)} \\right)$$\n\u5176\u4e2d${g'\\left( X \\right)}=\\frac{dg(x)}{dx}$.\u5176\u8bc1\u660e\u5982\u4e0b\uff1a\n$$\\begin{array}{l}\n{\\mathbb E}\\left( {g\\left( X \\right)\\left( {X - \\mu } \\right)} \\right) = \\int {g\\left( X \\right)\\left( {X - \\mu } \\right){\\cal N}\\left( {\\mu ,{\\sigma ^2}} \\right)dX} \\\\\n= \u00a0- \\frac{{{\\sigma ^2}}}{{\\sqrt {2\\pi } \\sigma }}\\int {g\\left( X \\right)d\\exp \\left( { - \\frac{{{{\\left( {X - \\mu } \\right)}^2}}}{{2{\\sigma ^2}}}} \\right)} \\\\\n= \\left. { - \\frac{{{\\sigma ^2}}}{{\\sqrt {2\\pi } \\sigma }}g\\left( X \\right)\\exp \\left( { - \\frac{{{{\\left( {X - \\mu } \\right)}^2}}}{{2{\\sigma ^2}}}} \\right)} \\right|_{ - \\infty }^{ + \\infty } + \\frac{{{\\sigma ^2}}}{{\\sqrt {2\\pi } \\sigma }}\\int {g'\\left( X \\right)\\exp \\left( { - \\frac{{{{\\left( {X - \\mu } \\right)}^2}}}{{2{\\sigma ^2}}}} \\right)} dX\\\\\n\\overset{a}{=} {\\sigma ^2}\\int {g'\\left( X \\right){\\cal N}\\left( {\\mu ,{\\sigma ^2}} \\right)} dX\\\\\n= {\\mathbb E}\\left( {g'\\left( X \\right)} \\right)\n\\end{array}$$\na\u5904\u7684\u7b49\u53f7\u7531\u6709\u754c\u6027\u5f97\u5230\u3002\u5728${\\bf{g}}\\left( {\\bf{X}} \\right):{{\\mathbb R}^M} \\to {{\\mathbb R}^N}$\uff0c${\\bf{X}}\\sim{\\cal N}\\left( {{\\bf{\\mu }},{\\bf{\\Sigma }}} \\right)$\u65f6\uff0c\u5176\u77e2\u91cf\u8868\u8fbe\u4e3a\n$${\\mathbb E}\\left( {\\left( {{\\bf{X}} - {\\bf{\\mu }}} \\right)} {\\bf{g}}{{\\left( {\\bf{X}} \\right)^T}}\\right) = {\\bf{\\Sigma }}{\\mathbb E}\\left( {\\nabla {\\bf{g}}{{\\left( {\\bf{X}} \\right)}}} \\right)$$\n\u5176\u4e2d$\\nabla \\left( \u00a0{\\bf g} \u00a0\\right) = \\left[ {\\begin{array}{}\n{\\frac{{\\partial {g_1}}}{{\\partial {x_1}}}}& \\cdots &{\\frac{{\\partial {g_1}}}{{\\partial {x_M}}}}\\\\\n\\vdots & \\ddots & \\vdots \\\\\n{\\frac{{\\partial {g_N}}}{{\\partial {x_1}}}}& \\cdots &{\\frac{{\\partial {g_N}}}{{\\partial {x_M}}}}\n\\end{array}} \\right]^T$. \u4e0b\u9762\u5bf9$g_i$\u5206\u91cf\u8fdb\u884c\u8bc1\u660e\n$$\\begin{array}{l}\n{\\mathbb E}\\left( {{g_i}\\left( {\\bf{X}} \\right)\\left( {{\\bf{X}} - {\\bf{\\mu }}} \\right)} \\right) = \\int {{g_i}\\left( {\\bf{X}} \\right)\\left( {{\\bf{X}} - {\\bf{\\mu }}} \\right){\\cal N}\\left( {{\\bf{\\mu }},{\\sigma ^2}} \\right)d{\\bf{X}}} \\\\\n= K{\\bf{\\Sigma }}\\int {{g_i}\\left( {\\bf{X}} \\right)d{\\cal N}\\left( {{\\bf{\\mu }},{\\sigma ^2}} \\right)} \\\\\n= \u00a0- K{\\bf{\\Sigma }}{g_i}\\left( {\\bf{X}} \\right)\\left( {{\\bf{X}} - {\\bf{\\mu }}} \\right){\\cal N}\\left.\\left( {{\\bf{\\mu }},{\\sigma ^2}} \\right)\\right|_{ - \\infty }^{ + \\infty } + K{\\bf{\\Sigma }}\\int {\\nabla {g_i}\\left( {\\bf{X}} \\right){\\cal N}\\left( {{\\bf{\\mu }},{\\sigma ^2}} \\right)d{\\bf{X}}} \\\\\n= {\\bf{\\Sigma }}{\\mathbb E}\\left( {\\nabla {g_i}\\left( {\\bf{X}} \\right)} \\right)\n\\end{array}$$\n\u6b64\u5904$\\nabla {g_i}\\left( {\\bf{X}} \\right) = {\\left[ {\\begin{array}{}\n{\\frac{{\\partial {g_i}\\left( {\\bf{X}} \\right)}}{{\\partial {x_j}}}}& \\cdots &{\\frac{{\\partial {g_i}\\left( {\\bf{X}} \\right)}}{{\\partial {x_M}}}}\n\\end{array}} \\right]^T}$\uff0c$K$\u4e3a\u5e38\u6570\uff08\u5982\u679c\u5bfc\u6570\u65b9\u5411\u4e0d\u4e00\u6837\u9700\u8981\u64cd\u4f5c\u4e0b\uff09\u3002\u9488\u5bf9\u8f93\u5165\u8f93\u51fa\u7b49\u957f\u7684\u60c5\u51b5\uff0c\u53d6\u5176\u8ff9\u53ef\u5f97\u5185\u79ef\u5f62\u5f0f\uff1a\n$${\\mathbb E}\\left( {{{\\left( {{\\bf{X}} - {\\bf{\\mu }}} \\right)}^T}{\\bf{g}}\\left( {\\bf{X}} \\right)} \\right) = \\sum\\limits_{j = 1}^N {\\sum\\limits_{k = 1}^N {\\sigma _{jk}^2{\\mathbb E}\\left( {\\frac{{\\partial {g_k}}}{{\\partial {x_j}}}} \\right)} }$$\n\u82e5\u534f\u65b9\u5dee\u4e3a\u5355\u4f4d\u9635\u7684\u6807\u91cf\u500d\uff08\u8fd9\u91cc\u4e3a${\\sigma ^2}$\uff09\uff0c\u5219\u8fd8\u53ef\u4ee5\u7ee7\u7eed\u7b80\u5316\u4e3a\u6563\u5ea6<\/strong>\n$${\\mathbb E}\\left( {{{\\left( {{\\bf{X}} - {\\bf{\\mu }}} \\right)}^T}{\\bf{g}}\\left( {\\bf{X}} \\right)} \\right) = {\\sigma ^2}{\\mathbb E}\\left(div\\left( {{\\bf{g}}\\left( {\\bf{X}} \\right)} \\right)\\right)$$<\/p>\n \u4ed4\u7ec6\u89c2\u5bdf\u8fd9\u4e2a\u5f15\u7406\uff0c\u82e5\u5c06${\\bf{g}}\\left( {\\bf{X}} \\right)$\u89c6\u4e3a\u67d0\u4e2a\u53bb\u566a\u5668\uff0c\u5176\u8f93\u51fa\u4fbf\u662f\u4f7f\u7528\u5148\u9a8c\u7ea6\u675f\u6216\u5176\u4ed6\u4e00\u4e9b\u4eba\u4e3a\u8bbe\u5b9a\u7684\u7ea6\u675f\u8fdb\u884c\u53bb\u566a\u7684\u7ed3\u679c\u3002\u800c\u8f93\u5165\u4e0e\u8f93\u51fa\u4e4b\u95f4\u7684\u663e\u6027\u76f8\u5173\u6027\uff0c\u4fbf\u7531\u8f93\u5165\u7684\u534f\u65b9\u5dee\u4ee5\u53ca\u53bb\u566a\u5668\u7684\u672c\u8eab\u7279\u6027\u51b3\u5b9a\uff0c\u8fd9\u662f\u975e\u5e38\u663e\u7136\u7684\u3002\u800c\u5728MMSE\u53bb\u566a\u5668\uff0c\u5373${\\bf{g}}\\left( {\\bf{r}} \\right) = {\\mathbb E}\\left( {{\\bf{x}}|{\\bf{r}}} \\right) = \\int {\\frac{1}{Z}{\\bf{x}}p\\left( {\\bf{x}} \\right){\\cal N}\\left( {{\\bf{x}};{\\bf{r}},\\sigma _r^2{\\bf{I}}} \\right)d{\\bf{x}}}$\u7684\u60c5\u51b5\u4e0b\uff0c\u5176\u4e2d${\\bf r}={\\bf x}+{\\bf n}_r$, ${\\bf n}_r\\sim{{\\cal N}\\left( {0 ,{\\sigma_r ^2}{\\bf I}} \\right)}$\uff0c\u8fd8\u6709\u66f4\u8fdb\u4e00\u6b65\u7684\u7ed3\u679c\uff1a\n$$\\begin{array}{l}\n\\frac{{\\partial {\\bf{g}}\\left( {\\bf{r}} \\right)}}{{\\partial {r_k}}} = \\int {\\frac{1}{Z}{\\bf{x}}p\\left( {\\bf{x}} \\right)\\frac{{\\partial {\\cal N}\\left( {{\\bf{x}};{\\bf{r}},\\sigma _r^2{\\bf{I}}} \\right)}}{{\\partial {r_k}}}d{\\bf{x}}} \\\\\n= \\int {\\frac{{{x_k} - {r_k}}}{{\\sigma _r^2}}\\frac{1}{Z}{\\bf{x}}p\\left( {\\bf{x}} \\right){\\cal N}\\left( {{\\bf{x}};{\\bf{r}},\\sigma _r^2{\\bf{I}}} \\right)d{\\bf{x}}} \u00a0+ \\int {\\frac{{\\partial \\frac{1}{Z}}}{{\\partial {r_k}}}{\\bf{x}}p\\left( {\\bf{x}} \\right){\\cal N}\\left( {{\\bf{x}};{\\bf{r}},\\sigma _r^2{\\bf{I}}} \\right)d{\\bf{x}}} \\\\\n= \\frac{1}{{\\sigma _r^2}}\\left( {{\\mathbb E}\\left( {{\\bf{x}}\\left( {{x_k} - {r_k}} \\right)|{\\bf{r}}} \\right) - {\\mathbb E}\\left( {{\\bf{x}}|{\\bf{r}}} \\right)\\left( {{\\mathbb E}\\left( {{x_k}|{\\bf{r}}} \\right) - {r_k}} \\right)} \\right)\\\\\n= \\frac{1}{{\\sigma _r^2}}\\left( {{\\mathbb E}\\left( {{\\bf{x}}{x_k}|{\\bf{r}}} \\right) - {\\mathbb E}\\left( {{\\bf{x}}|{\\bf{r}}} \\right){\\mathbb E}\\left( {{x_k}|{\\bf{r}}} \\right)} \\right)\n\\end{array}$$\n\u5176\u4e2d\n$$\\begin{array}{l}\n\\frac{{\\partial \\frac{1}{Z}}}{{\\partial {r_k}}} = \u00a0- \\frac{1}{{{Z^2}}}\\int {p\\left( {\\bf{x}} \\right)\\frac{{\\partial {\\cal N}\\left( {{\\bf{x}};{\\bf{r}},\\sigma _r^2{\\bf{I}}} \\right)}}{{\\partial {r_k}}}d{\\bf{x}}} \\\\\n= \u00a0- \\frac{1}{{{Z^2}}}\\int {p\\left( {\\bf{x}} \\right)\\frac{{{x_k} - {r_k}}}{{\\sigma _r^2}}{\\cal N}\\left( {{\\bf{x}};{\\bf{r}},\\sigma _r^2{\\bf{I}}} \\right)d{\\bf{x}}} \\\\\n= \u00a0- \\frac{1}{{\\sigma _r^2Z}}\\left( {{\\mathbb E}\\left( {{x_k}|{\\bf{r}}} \\right) - {r_k}} \\right)\n\\end{array}$$\n\u800c\u5199\u6210\u77e2\u91cf\u5f62\u5f0f\u540e\uff0c\u6709\n$$\\frac{{\\partial {\\bf{g}}\\left( {\\bf{r}} \\right)}}{{\\partial {\\bf{r}}}} = \\frac{1}{{\\sigma _r^2}}\\left( {{\\mathbb E}\\left( {{\\bf{x}}{{\\bf{x}}^T}|{\\bf{r}}} \\right) - {\\mathbb E}\\left( {{\\bf{x}}|{\\bf{r}}} \\right){\\mathbb E}\\left( {{{\\bf{x}}^T}|{\\bf{r}}} \\right)} \\right)$$\n\u8fd9\u6b63\u662fMMSE\u4f30\u8ba1\u7684\u534f\u65b9\u5dee\u77e9\u9635\u3002\u82e5\u53d6\u6563\u5ea6\uff0c\u5219\u5f97\u5230\n$$div\\left( {{\\bf{g}}\\left( {\\bf{r}} \\right)} \\right) = \\frac{1}{{\\sigma _r^2}}\\sum\\limits_{i = 1}^N {Var\\left( {{x_i}|{\\bf{r}}} \\right)}$$\n\u8fd9\u4e0eOAMP\/VAMP\u7684\u7ed3\u8bba\u76f8\u7b26\u3002\n\u5b9e\u9645\u4e0a\uff0c\u4ece\u6d88\u606f\u4f20\u9012\u7684\u89d2\u5ea6\u4e0a\u770b\uff0cAMP\/OAMP\/VAMP\u4e2d\u7684Onsager\u9879(\u8865\u507f)\uff0c\u6765\u81ea\u4e8e\u6cf0\u52d2\u5c55\u5f00\u7684\u4e00\u9636\u5bfc\u6570\uff08\u4e4b\u524d\u535a\u5ba2\u91cc\u9762\u6709\uff09\uff0c\u800c\u5728\u5f88\u7279\u6b8a\u7684\u4f30\u8ba1\u5668\uff08\u8fd9\u91cc\u662fMMSE\u4f30\u8ba1\u5668\uff09\u4e0b\uff0c\u5176\u4f1a\u9000\u5316\u4e3a\u65b9\u5dee\uff0c\u8fd9\u624d\u6709\u5e38\u89c1\u7684\u95ed\u5f62\u5f0f\u7684AMP\/OAMP\/VAMP\u7684Onsager\u9879\u3002\u800c\u4eceStein's lemma\u7684\u89d2\u5ea6\u770b\uff0cOnsager\u9879\u4e2d\u7684\u4e00\u9636\u5bfc\u6570\uff0c\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u201c\u76f8\u5173\u201d\u9879\uff0c\u5176\u63cf\u8ff0\u8f93\u51fa<\/strong>\u4e0e\u8f93\u5165\u566a\u58f0<\/strong>\u7684\u76f8\u5173\u6027\uff0c\u800c\u540e\u5728AMP\/OAMP\/VAMP\u4e2d\u5c06\u5176\u53bb\u6389\uff0c\u4ee5\u83b7\u5f97\u66f4\u597d\u7684\u4f30\u8ba1\u6548\u679c\u3002<\/p>\n \u590d\u6570\u60c5\u51b5\u4e0b\u6709\u7a7a\u518d\u5199\u3002\u3002\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u56fd\u5e86\u6ca1\u4eba\u627e\u6211\u73a9\uff0c\u597d\u65e0\u804a\uff0c\u501f\u6b64\u4e5f\u68b3\u7406\u4e00\u4e0b\u5173\u4e8eStein’s lemma\u7684\u76f8\u5173\u5185\u5bb9\u3002 Stein’s lemma\uff08\u65af\u5766\u5f15\u7406\uff09 \u9996\u5148\uff0c\u4ecb\u7ecd\u4e00\u4e0b\u8fd9\u4e2a\u5f15\u7406\u3002\u5355\u53d8\u91cf\u7684\u60c5\u51b5\u4e0b\uff0c$X\\sim{\\cal N}\\left( {\\mu ,{\\sigma ^2}} \\right)$\uff0c\u4e14$g\\left( X \\right):{{\\mathbb R}^1} \\to {{\\mathbb R}^1}$\uff0c${\\mathbb E}\\left( {g\\left( X \\right)} \\right)$\u4ee5\u53ca${\\mathbb E}\\left( {g’\\left( X \\right)} \\right)$\u4e3a\u6709\u754c\uff0c\u5219 $${\\mathbb…… <\/p>\n \u7ee7\u7eed\u9605\u8bfb»»»<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"_links":{"self":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/posts\/105"}],"collection":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=105"}],"version-history":[{"count":0,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/posts\/105\/revisions"}],"wp:attachment":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=105"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=105"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=105"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}MMSE\u4f30\u8ba1\u4e2d\u7684\u7ed3\u8bba<\/h1>\n