{"id":831,"date":"2017-03-06T20:42:38","date_gmt":"2017-03-06T12:42:38","guid":{"rendered":"http:\/\/planckscale.info\/?p=831"},"modified":"2017-03-07T09:41:24","modified_gmt":"2017-03-07T01:41:24","slug":"%e7%bb%9f%e8%ae%a1%e7%ae%97%e6%b3%95%e7%9a%84%e7%90%86%e8%ae%ba%e4%b8%8e%e5%ae%9e%e4%be%8b%e4%b9%8b%e4%b8%80-%e6%9c%80%e5%a4%a7%e4%bc%bc%e7%84%b6%e4%bc%b0%e8%ae%a1%e4%b8%8eem%e7%ae%97%e6%b3%95","status":"publish","type":"post","link":"https:\/\/mindspectrum.xyz\/en\/2017\/03\/06\/%e7%bb%9f%e8%ae%a1%e7%ae%97%e6%b3%95%e7%9a%84%e7%90%86%e8%ae%ba%e4%b8%8e%e5%ae%9e%e4%be%8b%e4%b9%8b%e4%b8%80-%e6%9c%80%e5%a4%a7%e4%bc%bc%e7%84%b6%e4%bc%b0%e8%ae%a1%e4%b8%8eem%e7%ae%97%e6%b3%95\/","title":{"rendered":"\u7edf\u8ba1\u7b97\u6cd5\u7684\u7406\u8bba\u4e0e\u5b9e\u4f8b\u4e4b\u4e00 \u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e0eEM\u7b97\u6cd5"},"content":{"rendered":"

\u7248\u6743\u58f0\u660e\uff1a\u539f\u521b\u4f5c\u54c1\uff0c\u6b22\u8fce\u8f6c\u8f7d\uff0c\u4f46\u8f6c\u8f7d\u8bf7\u4ee5\u8d85\u94fe\u63a5\u5f62\u5f0f\u6ce8\u660e\u6587\u7ae0\u6765\u6e90(planckscale.info)\u3001\u4f5c\u8005\u4fe1\u606f\u548c\u672c\u58f0\u660e\uff0c\u5426\u5219\u5c06\u8ffd\u7a76\u6cd5\u5f8b\u8d23\u4efb\u3002<\/strong><\/p>\n

\u53bb\u5e74\u548c\u670b\u53cb\u505a\u4e86\u4e00\u4e9b\u7edf\u8ba1\u65b9\u9762\u7684\u4e1c\u897f\uff0c\u4e00\u4e9b\u7b14\u8bb0\u53d1\u5728\u8fd9\u513f\u3002\u7edf\u8ba1\u6211\u662f\u5916\u884c\uff0c\u8d3b\u7b11\u5927\u65b9\u4e4b\u5904\u8bf7\u6307\u6b63\u3002<\/p>\n

\\section{\u8bb0\u53f7\u7ea6\u5b9a}
\n\u672c\u6587\u4e2d\u7528\u5927\u5199\u5b57\u6bcd\"X,Y,Z...\"\u8868\u793a\u968f\u673a\u53d8\u91cf\uff0c\u5c0f\u5199\u5b57\u6bcd\"x,y,z...\"\u8868\u793a\u968f\u673a\u53d8\u91cf\u7684\u89c2\u6d4b\u503c\uff0c\u7c97\u4f53\u8868\u793a\u5411\u91cf\/\u77e9\u9635\u3002\"\mathcal{X},\mathcal{Y},\mathcal{Z}...\"\u8868\u793a\u5411\u91cf\u503c\u968f\u673a\u53d8\u91cf\u6216\u968f\u673a\u53d8\u91cf\"X,Y,Z...\"\u7684\u89c2\u6d4b\u6837\u672c\u3002<\/p>\n

\\section{\u4f30\u8ba1\u7406\u8bba}
\n\u7ed9\u968f\u673a\u53d8\u91cf\u4e00\u4e2a\u5f85\u5b9a\u53c2\u6570\u7684\u6982\u7387\u6a21\u578b\uff0c\u901a\u8fc7\u4e00\u7cfb\u5217\u6837\u672c\u89c2\u6d4b\uff0c\u8403\u53d6\u51fa\u6982\u7387\u6a21\u578b\u6700\u53ef\u80fd\u53c2\u6570\u7684\u7406\u8bba\u3002<\/p>\n

\\subsection{\u4e00\u4e9b\u6570\u5b66}
\n\u8003\u8651\u968f\u673a\u53d8\u91cf\"X,Y\"\u7684\u6982\u7387\u6a21\u578b\"p(X,Y)\",\u6211\u4eec\u7ecf\u5e38\u9700\u8981\u8003\u8651\u4e00\u7cfb\u5217\u89c2\u6d4b\u6837\u672c\"\mathcal{X}=\{x_1,...x_N\}\",\"\mathcal{Y}=\{y_1,...y_N\}\"\u7684\u5206\u5e03<\/p>\n

(1) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u89c2\u6d4b\u6837\u672c\u4e4b\u95f4\u7684\u72ec\u7acb\u6027\u610f\u5473\u7740\u8fb9\u7f18\u5206\u5e03\"p(\mathcal{X})\"\uff08\u6216\"p(\mathcal{Y})\"\uff09\uff0c\u6761\u4ef6\u5206\u5e03\"p(\mathcal{X}\vert\mathcal{Y})\"\uff08\u6216\"p(\mathcal{Y}\vert\mathcal{X})\"\uff09\u4e5f\u6ee1\u8db3\u7c7b\u4f3c\u7684\u56e0\u5b50\u5206\u89e3\u6027\u8d28\u3002\u4e25\u683c\u8bf4\u6765\uff0c\u5047\u8bbe\"Y\"\u53d6\u503c\u4e8e\"\Delta\",\u5219\"\mathcal{Y}\"\u53d6\u503c\u4e8e\"\otimes\Delta^N\",\u6211\u4eec\u6709<\/p>\n

(2) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\subsection{\u6700\u5927\u4f3c\u7136\u4f30\u8ba1}
\n\u5047\u8bbe\u968f\u673a\u53d8\u91cf\"X\"\u7684\u6982\u7387\u6a21\u578b\u4e3a\"p(X\vert\theta)\",\u5176\u4e2d\"\theta\"\u4e3a\u6a21\u578b\u53c2\u6570\u3002\u6211\u4eec\u6709\u4e00\u7cfb\u5217\u72ec\u7acb\u89c2\u6d4b\u6837\u672c\"\mathcal{X}=\{x_1,...x_N\}\",\u6b32\u5b9a\u51fa\u6700\u53ef\u80fd\u7684\u6a21\u578b\u53c2\u6570\"\theta_{\star}\".<\/p>\n

\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u5373\u5047\u8bbe\u89c2\u6d4b\u7ed3\u679c\u662f\u6700\u5927\u53ef\u80fd\u7684\u7ed3\u679c\uff0c\u4ece\u800c\u5b9a\u51fa\u5206\u5e03\u53c2\u6570\u7684\u4f30\u8ba1\u65b9\u6cd5\u3002\u5b9a\u4e49\u5bf9\u6570\u4f3c\u7136\u51fd\u6570<\/p>\n

(3) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5219\u6a21\u578b\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e3a<\/p>\n

(4) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u6ce8\u610f\"-\log p(x_i,\theta)\"\u662f\u7b2c\"i\"\u6b21\u89c2\u6d4b\u6240\u83b7\u53d6\u5230\u7684\u4fe1\u606f\u91cf\uff0c\u6b64\u76ee\u6807\u51fd\u6570\u4e00\u4e2a\u542b\u4e49\u662f\\textbf{\u8c03\u6574\u53c2\u6570\u4f7f\u5f97\u89c2\u6d4b\u5230\u7684\u4fe1\u606f\u91cf\u53d6\u6781\u5c0f}\u3002<\/p>\n

\\subsection{\u6982\u7387\u5206\u5e03\u7684\u63cf\u8ff0\u80fd\u529b}
\n\u4f30\u8ba1\u7406\u8bba\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u4ee5\u4f30\u8ba1\u5206\u5e03\u7684\u5f62\u5f0f\u5f97\u5230\u6837\u672c\u7a7a\u95f4\u4e2d\u7684\u51e0\u4f55\u5bf9\u8c61\u3002\u6982\u7387\u5206\u5e03\u7684\u63cf\u8ff0\u80fd\u529b\u662f\u5e7f\u6cdb\u7684\uff0c\u8fd9\u91cc\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002<\/p>\n

\\textbf{\u7b80\u5355\u692d\u7403\u533a\u57df}\u3002\u53ef\u4ee5\u7b80\u5355\u7684\u7531Gauss\u5206\u5e03\"p(\boldsymbol{X})=\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})\"\u523b\u753b\uff0c\"\boldsymbol{\mu}\"\u7ed9\u51fa\u4e86\u4f4d\u7f6e\u800c\"\boldsymbol{\Sigma}\"\u5219\u7ed9\u51fa\u4e86\u692d\u7403\u5728\u5404\u65b9\u5411\u4e0a\u7684\u8f74\u957f\u3002<\/p>\n

\\textbf{\u591a\u4e2a\u692d\u7403\u533a\u57df}\u3002\u7531\u591a\u4e2aGauss\u5206\u5e03\u7684\u53e0\u52a0\uff0c\u5373\\textbf{\u6df7\u5408Gauss\u5206\u5e03}\u63cf\u8ff0\u3002\u6982\u7387\u5bc6\u5ea6\u5206\u5e03\"\mathcal{M}(\{ \boldsymbol{\mu}_a,\boldsymbol{\Sigma}_a,\boldsymbol{\pi}_a\})\"\u4e3a<\/p>\n

(5) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5206\u5e03\u63cf\u8ff0\u4e86\u6837\u672c\u7a7a\u95f4\u91cc\u591a\u4e2a\u7b80\u5355\u692d\u7403\u533a\u57df\uff0c\u6bcf\u4e2a\u533a\u57df\u53ef\u4ee5\u4ee3\u8868\u4e00\u4e2a\u6570\u636e\u7c7b\u3002\u901a\u5e38\u5728\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u628a\u89c2\u6d4b\u4e0d\u5230\u7684\u9690\u542b\u53d8\u91cf\uff0c\u7c7b\u522b\u6807\u7b7e\"Z\"\u52a0\u5165\u5230\u7edf\u8ba1\u91cf\u96c6\u5408\u4e2d\uff0c\u4e8e\u662f\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u6210\u4e3a<\/p>\n

(6) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\textbf{\u51fd\u6570\u56fe\u5f62}\u3002\u51fd\u6570\"f(x)\"\u5728\u6837\u672c\u7a7a\u95f4\u91cc\u7684\u56fe\u5f62\u53ef\u4ee5\u7531\u6982\u7387\u5206\u5e03<\/p>\n

(7) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u523b\u753b\u3002\u7ed9\u5b9a\u4efb\u610f\"X\",\"Y\"\u90fd\u6ee1\u8db3\u4e00\u4e2a\u4ee5\"f(X)\"\u4e3a\u4e2d\u5fc3\u7684Gauss\u5206\u5e03\uff0c\u56e0\u800c\u8fd9\u91cc\u523b\u753b\u4e86\u4e00\u4e2a\u5bf9\"Y=f(X)\"\u8fd9\u4e00\u80cc\u540e\u672c\u8d28\u7684\u6570\u91cf\u5173\u7cfb\u6240\u505a\u7684\u6d4b\u91cf\uff0c\u800c\u8be5\u6d4b\u91cf\u662f\u5e26\u6709Gauss\u566a\u58f0\u7684\u3002<\/p>\n

\\subsection{\u6700\u5c0f\u4e8c\u4e58\u6cd5}
\n\u8003\u8651\u51fd\u6570\u65cf\"f_{\boldsymbol{\alpha}}(x)\",\u6211\u4eec\u901a\u8fc7\u4e00\u7cfb\u5217\u6d4b\u91cf\u6570\u636e\"\mathcal{X}\equiv\{x_{1},...x_N\},\mathcal{Y}\equiv\{y_{1},...y_N\}\"\u53bb\u4f30\u8ba1\u80cc\u540e\u771f\u5b9e\u7684\u51fd\u6570\u5173\u7cfb\"f_*(x)\".
\n\u5047\u8bbe\u89c2\u6d4b\u566a\u58f0\u4e3aGauss\u5206\u5e03\uff0c\u4e8e\u662f\u89c2\u6d4b\u7684\u6982\u7387\u5206\u5e03\u4e3a<\/p>\n

(8) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u6784\u9020\u5bf9\u6570\u4f3c\u7136\u51fd\u6570<\/p>\n

(9) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u4e8e\u662f\uff0c\"\max_{\theta} \log L(\theta;\mathcal{X},\mathcal{Y})\"\u7ed9\u51fa<\/p>\n

(10) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d\"\sigma=\sqrt{\sum_i(y_i-f_{*}(x_i))^2/N}\".<\/p>\n

\u4e8e\u662f\u6211\u4eec\u5f97\u5230\u4e86\u6700\u5c0f\u4e8c\u4e58\u6cd5\u3002<\/p>\n

\\subsection{EM\u7b97\u6cd5}
\n\u5bf9\u4e8e\u4e00\u4e2a\u8054\u5408\u5206\u5e03\uff0c\u5b9e\u9645\u5e94\u7528\u4e2d\u5e76\u975e\u6bcf\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u90fd\u53ef\u4ee5\u88ab\u89c2\u6d4b\u5230\u3002\u6bd4\u5982\u4e0a\u8ff0\u6df7\u5408Gauss\u5206\u5e03\u4e2d\uff0c\u901a\u5e38\u7c7b\u522b\u6807\u7b7e\"Z\"\u662f\u65e0\u6cd5\u89c2\u6d4b\u5230\u7684\u3002\u8fd9\u79cd\u4e0d\u5b8c\u5907\u89c2\u6d4b\u60c5\u51b5\u4e0b\u76f4\u63a5\u4f7f\u7528\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4f1a\u51fa\u73b0\u8ba1\u7b97\u91cf\u8fc7\u5927\uff0c\u5b9e\u9645\u4e0d\u53ef\u89e3\u7684\u95ee\u9898\u3002\u4e8e\u662f\u5bf9\u4e8e\u8fd9\u7c7b\u4f30\u8ba1\u95ee\u9898\uff0c\u6211\u4eec\u4f7f\u7528\\textbf{EM\u7b97\u6cd5}\u6765\u6c42\u89e3\u3002<\/p>\n

\u8003\u8651\u6df7\u5408Gauss\u5206\u5e03<\/p>\n

(11) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u8bbe\"\mathcal{X}\equiv\{x_{1},...x_N\}\"\u4e3a\u89c2\u6d4b\u6837\u672c,\"\mathcal{Z}\equiv\{z_1,...z_N\}\"\u4e3a\u89c2\u6d4b\u6837\u672c\u5bf9\u5e94\u7684\u9690\u53d8\u91cf\u3002\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u4e3a<\/p>\n

(12) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d\u8fd9\u4e00\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u5728\u8ba1\u7b97\u4e0a\u6709\u8bf8\u591a\u4e0d\u4fbf\uff0c\u9996\u5148\u52a0\u548c\u53f7\u963b\u6b62\u4e86\u5bf9\u6570\u8fd0\u7b97\u4f5c\u7528\u5728\"p\"\u4e0a\uff0c\u518d\u8005\u5728\"z\"\u53d8\u91cf\u53d6\u503c\u8fd4\u56de\u8f83\u5e7f\u65f6\u8ba1\u7b97\u4ee3\u4ef7\u4e5f\u4f1a\u5f88\u5927\u3002\u56e0\u6b64\u4f20\u7edf\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u5728\u6b64\u5904\u4e0d\u9002\u7528\u3002<\/p>\n

\u5728EM\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u5728\u6bcf\u4e00\u4e2a\u8fed\u4ee3\u6b65\u90fd\u6784\u9020\u4e00\u4e2a\u65b0\u7684\u51fd\u6570\"Q^{(t)}\",\u7528\"\max Q^{(t)}\"\u6765\u8fd1\u4f3c\"\log L(\boldsymbol{\theta};\mathcal{X})\"\u5728\u5f53\u524d\u89e3\"\boldsymbol{\theta}^{(t)}\"\u9644\u8fd1\u90bb\u57df\u4e0a\u7684\u53d8\u5316\u89c4\u5f8b,\u8fed\u4ee3\u6536\u655b\u5230\u6700\u7ec8\u89e3<\/p>\n

(13) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d<\/p>\n

(14) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u8be5\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u8bc1\u660e\u7559\u5230\u4e0b\u4e00\u5c0f\u8282\u5904\u7406\u3002<\/p>\n

\\subsection{EM\u7b97\u6cd5\u7684\u7406\u8bba\u57fa\u7840}
\n\u5728\u76ee\u6807\u51fd\u6570\u8f83\u4e3a\u590d\u6742\u7684\u60c5\u51b5\u4e0b\uff0c\u6781\u5927\/\u6781\u5c0f\u503c\u7684\u5bfb\u627e\u901a\u5e38\u91c7\u7528\u8fed\u4ee3\u7b97\u6cd5\u6765\u5904\u7406\u3002\u8ba1\u7b97\u76ee\u6807\u51fd\u6570\u7cbe\u786e\u7684\u5f62\u72b6\u662f\u56f0\u96be\u7684\uff0c\u56e0\u800c\u6211\u4eec\u9700\u8981\\textbf{\u5728\u6bcf\u4e00\u8fed\u4ee3\u6b65\u901a\u8fc7\u6709\u9650\u7684\u8ba1\u7b97\uff0c\u63a2\u77e5\u76ee\u6807\u51fd\u6570\u5728\u5f53\u524d\u8fed\u4ee3\u89e3\u4e4b\u90bb\u57df\u4e0a\u7684\u53d8\u5316\u89c4\u5f8b\uff0c\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u51fd\u6570\u6765\u523b\u753b\u8fd9\u4e00\u5c40\u57df\u53d8\u5316\u8d8b\u52bf\uff0c\u7136\u540e\u6c42\u89e3\u8fd9\u4e00\u5c40\u90e8\u7684\u6781\u5c0f\/\u6781\u5927\u5316\u95ee\u9898\uff0c\u6700\u540e\u8fdb\u5165\u4e0b\u4e00\u8f6e\u8fed\u4ee3}\u3002\u8fd9\u6837\uff0c\u4e00\u4e2a\u590d\u6742\u76ee\u6807\u51fd\u6570\u7684\u6781\u5c0f\/\u6781\u5927\u95ee\u9898\u5c31\u88ab\u5206\u89e3\u6210\u4e86\u4e00\u7cfb\u5217\u5bb9\u6613\u6c42\u89e3\u7684\u7b80\u5355\u95ee\u9898\uff0c\u901a\u8fc7\u8fed\u4ee3\u4e0d\u65ad\u9760\u8fd1\u6781\u4f18\u89e3\u3002<\/p>\n

\u5177\u4f53\u8bf4\u6765\uff0c\u4e0d\u540c\u7684\u7b97\u6cd5\u4f1a\u4e3a\u76ee\u6807\u51fd\u6570\u7684\u5c40\u57df\u53d8\u5316\u5efa\u7acb\u4e0d\u540c\u7684\u6a21\u578b\uff0c\u6bd4\u5982\u901a\u8fc7\u8ba1\u7b97\u5bfc\u6570\uff0c\u7528\u5e73\u9762\u6765\u5efa\u6a21\u8be5\u5904\u7684\u76ee\u6807\u51fd\u6570\u7684\u53d8\u5316\uff1b\u518d\u5982\u8ba1\u7b97\u5230\u4e8c\u9636\u5bfc\u6570\uff0c\u7528\u629b\u7269\u9762\u6765\u8fd1\u4f3c\u76ee\u6807\u51fd\u6570\uff0c\u7b49\u7b49\u3002<\/p>\n

EM\u7b97\u6cd5\u4e5f\u662f\u8fd9\u6837\u4e00\u79cd\u7b97\u6cd5\u3002\\textbf{\u5b83\u7684\u7279\u6b8a\u4e4b\u5904\u5728\u4e8e\uff0c\u7531\u4e8e\u76ee\u6807\u51fd\u6570\uff0c\u5373\u6982\u7136\u51fd\u6570\u7684\u7279\u6b8a\u7ed3\u6784\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u6bd4\u4e00\u6b21\/\u4e8c\u6b21\u8fd1\u4f3c\u66f4\u597d\u7684\u5c40\u57df\u53d8\u5316\u6a21\u578b}\uff0c\u4ece\u800c\u83b7\u5f97\u6bd4\u5e38\u89c4\u7b97\u6cd5\u66f4\u597d\u7684\u6027\u80fd\u3002\u8fd9\u4e00\u5c40\u57df\u53d8\u5316\u6a21\u578b\u5c31\u662f\"Q^{(t)}(\boldsymbol{\theta})\".<\/p>\n

\u9996\u5148\u6211\u4eec\u6765\u5bfb\u627e\"Q^{(t)}(\boldsymbol{\theta})\"\u548c\"\log p(\mathcal{X}\vert \boldsymbol{\theta})\"\u4e4b\u95f4\u7684\u5173\u7cfb
\n<\/a><\/p>\n

(15) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u6613\u8bc1\"H(\boldsymbol{\theta} \vert \boldsymbol{\theta}^{(t)})\geq 0\",\u53ef\u89c1<\/p>\n

(16) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5373\\textbf{\"Q(\boldsymbol{\theta} \vert \boldsymbol{\theta}^{(t)})\"\u6784\u6210\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u7684\u4e0b\u754c}\u3002\u5c0615<\/a>\u51cf\u53bb\u5176\u5728\"\boldsymbol{\theta}^{(t)}\"\u5904\u7684\u503c\uff0c\u53c8\u6709
\n<\/a><\/p>\n

(17) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5373\\textbf{\u6bcf\u4e00\u8fed\u4ee3\u6b65\u4e2d\u5fc5\u6709\"\Delta \log L \geq \Delta Q\"}.<\/p>\n

\u53ef\u89c1\"Q^{(t)}(\boldsymbol{\theta})\"\u53ef\u4ee5\u53cd\u6620\"\log p(\mathcal{X}\vert \boldsymbol{\theta})\"\u5728\"\boldsymbol{\theta}^{(t)}\"\u5904\u7684\u53d8\u5316\u89c4\u5f8b\u3002\u66f4\u91cd\u8981\u7684\u662f\uff0c\"Q^{(t)}(\boldsymbol{\theta})\"\u6784\u6210\u4e86\u5bf9\u6570\u6982\u7136\u51fd\u6570\u7684\u4e00\u4e2a\u4e0b\u754c\uff0c\u8fd9\u6837\u4ee5\u6765\u6211\u4eec\u5c31\u53ef\u4ee5\u653e\u5fc3\u7684\u6c42\u89e3\"\max Q^{(t)}(\boldsymbol{\theta})\"\u8fd9\u4e00\u95ee\u9898\u6765\u5f97\u5230\u4e00\u4e2a\u66f4\u597d\u7684\u8fed\u4ee3\u89e3\uff0c\u800c\u4e0d\u7528\u50cfLM\u7b97\u6cd5\u4e2d\u90a3\u6837\u62c5\u5fc3\u4e8c\u9636\u6a21\u578b\u53ea\u5728\u4e00\u4e2a\u5c40\u57df\u7684\u8303\u56f4\u5185\u6784\u6210\u76ee\u6807\u51fd\u6570\u7684\u6709\u6548\u8fd1\u4f3c\uff0c\u9700\u8981\u7528\u7f6e\u4fe1\u57df\u65b9\u6cd5\u907f\u514d\u5927\u6b65\u957f\u7684\u4f18\u5316\u3002<\/p>\n

\u7b97\u6cd5\u7684\u6536\u655b\u6027\u7531\u8fd9\u4e00\u70b9\u6765\u4fdd\u8bc1\u3002\u7531\u4e8e\"D(p(\mathcal{Z}\vert \mathcal{X},\boldsymbol{\theta}^{(t)}) \mid p(\mathcal{Z}\vert \mathcal{X},\boldsymbol{\theta}))\"\u5728\"\boldsymbol{\theta}=\boldsymbol{\theta}^{(t)}\"\u9644\u8fd1\u7684\u4e00\u9636\u5c55\u5f00\u9879\u4e3a\u96f6\uff0c\u4e8e\u662f17<\/a>\u7684\u4e00\u9636\u5c55\u5f00\u7ed9\u51fa
\n<\/a><\/p>\n

(18) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u7b97\u6cd5\u8fed\u4ee3\u5230\"Q\"\u4e0d\u518d\u6709\u660e\u663e\u53d8\u5316(\"\boldsymbol{\theta}^{(t+1)}\approx\boldsymbol{\theta}^{(t)}\")\u65f6\uff0c\u4e0a\u5f0f\u53f3\u7aef\u4e3a\u96f6\uff0c\u5f0f18<\/a>\u610f\u5473\u7740\"\boldsymbol{\theta}=\boldsymbol{\theta}^{(t)}\"\u540c\u6837\u4e5f\u662f\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u7684\u6781\u503c\u70b9\u3002\u4e8e\u662f\"Q\"\u7684\u6536\u655b\u610f\u5473\u7740\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u7684\u6536\u655b\uff0c\u7b97\u6cd5\u5fc5\u7136\u6536\u655b\u4e8e\u4e00\u4e2a\u6781\u4f18\u89e3\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u7248\u6743\u58f0\u660e\uff1a\u539f\u521b\u4f5c\u54c1\uff0c\u6b22\u8fce\u8f6c\u8f7d\uff0c\u4f46\u8f6c\u8f7d\u8bf7\u4ee5\u8d85\u94fe\u63a5\u5f62\u5f0f\u6ce8\u660e\u6587\u7ae0\u6765\u6e90(planckscale.info)\u3001\u4f5c\u8005\u4fe1\u606f\u548c\u672c\u58f0\u660e\uff0c\u5426\u5219\u5c06\u8ffd\u7a76\u6cd5\u5f8b\u8d23\u4efb\u3002 \u53bb\u5e74\u548c\u670b\u53cb\u505a\u4e86\u4e00\u4e9b\u7edf\u8ba1\u65b9\u9762\u7684\u4e1c\u897f\uff0c\u4e00\u4e9b\u7b14\u8bb0\u53d1\u5728\u8fd9\u513f\u3002\u7edf\u8ba1\u6211\u662f\u5916\u884c\uff0c\u8d3b\u7b11\u5927\u65b9\u4e4b\u5904\u8bf7\u6307\u6b63\u3002 \\section{\u8bb0\u53f7\u7ea6\u5b9a} \u672c\u6587\u4e2d\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\u968f\u673a\u53d8\u91cf\uff0c\u5c0f\u5199\u5b57\u6bcd\u8868\u793a\u968f\u673a\u53d8\u91cf\u7684\u89c2\u6d4b\u503c\uff0c\u7c97\u4f53\u8868\u793a\u5411\u91cf\/\u77e9\u9635\u3002\u8868\u793a\u5411\u91cf\u503c\u968f\u673a\u53d8\u91cf\u6216\u968f\u673a\u53d8\u91cf\u7684\u89c2\u6d4b\u6837\u672c\u3002 \\section{\u4f30\u8ba1\u7406\u8bba} \u7ed9\u968f\u673a\u53d8\u91cf\u4e00\u4e2a\u5f85\u5b9a\u53c2\u6570\u7684\u6982\u7387\u6a21\u578b\uff0c\u901a\u8fc7\u4e00\u7cfb\u5217\u6837\u672c\u89c2\u6d4b\uff0c\u8403\u53d6\u51fa\u6982\u7387\u6a21\u578b\u6700\u53ef\u80fd\u53c2\u6570\u7684\u7406\u8bba\u3002 \\subsection{\u4e00\u4e9b\u6570\u5b66} \u8003\u8651\u968f\u673a\u53d8\u91cf\u7684\u6982\u7387\u6a21\u578b,\u6211\u4eec\u7ecf\u5e38\u9700\u8981\u8003\u8651\u4e00\u7cfb\u5217\u89c2\u6d4b\u6837\u672c,\u7684\u5206\u5e03 (1)   \u89c2\u6d4b\u6837\u672c\u4e4b\u95f4\u7684\u72ec\u7acb\u6027\u610f\u5473\u7740\u8fb9\u7f18\u5206\u5e03\uff08\u6216\uff09\uff0c\u6761\u4ef6\u5206\u5e03\uff08\u6216\uff09\u4e5f\u6ee1\u8db3\u7c7b\u4f3c\u7684\u56e0\u5b50\u5206\u89e3\u6027\u8d28\u3002\u4e25\u683c\u8bf4\u6765\uff0c\u5047\u8bbe\u53d6\u503c\u4e8e,\u5219\u53d6\u503c\u4e8e,\u6211\u4eec\u6709 (2)   \\subsection{\u6700\u5927\u4f3c\u7136\u4f30\u8ba1} \u5047\u8bbe\u968f\u673a\u53d8\u91cf\u7684\u6982\u7387\u6a21\u578b\u4e3a,\u5176\u4e2d\u4e3a\u6a21\u578b\u53c2\u6570\u3002\u6211\u4eec\u6709\u4e00\u7cfb\u5217\u72ec\u7acb\u89c2\u6d4b\u6837\u672c,\u6b32\u5b9a\u51fa\u6700\u53ef\u80fd\u7684\u6a21\u578b\u53c2\u6570. \u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u5373\u5047\u8bbe\u89c2\u6d4b\u7ed3\u679c\u662f\u6700\u5927\u53ef\u80fd\u7684\u7ed3\u679c\uff0c\u4ece\u800c\u5b9a\u51fa\u5206\u5e03\u53c2\u6570\u7684\u4f30\u8ba1\u65b9\u6cd5\u3002\u5b9a\u4e49\u5bf9\u6570\u4f3c\u7136\u51fd\u6570 (3)   \u5219\u6a21\u578b\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e3a (4)   \u6ce8\u610f\u662f\u7b2c\u6b21\u89c2\u6d4b\u6240\u83b7\u53d6\u5230\u7684\u4fe1\u606f\u91cf\uff0c\u6b64\u76ee\u6807\u51fd\u6570\u4e00\u4e2a\u542b\u4e49\u662f\\textbf{\u8c03\u6574\u53c2\u6570\u4f7f\u5f97\u89c2\u6d4b\u5230\u7684\u4fe1\u606f\u91cf\u53d6\u6781\u5c0f}\u3002 \\subsection{\u6982\u7387\u5206\u5e03\u7684\u63cf\u8ff0\u80fd\u529b} \u4f30\u8ba1\u7406\u8bba\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u4ee5\u4f30\u8ba1\u5206\u5e03\u7684\u5f62\u5f0f\u5f97\u5230\u6837\u672c\u7a7a\u95f4\u4e2d\u7684\u51e0\u4f55\u5bf9\u8c61\u3002\u6982\u7387\u5206\u5e03\u7684\u63cf\u8ff0\u80fd\u529b\u662f\u5e7f\u6cdb\u7684\uff0c\u8fd9\u91cc\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002 \\textbf{\u7b80\u5355\u692d\u7403\u533a\u57df}\u3002\u53ef\u4ee5\u7b80\u5355\u7684\u7531Gauss\u5206\u5e03\u523b\u753b\uff0c\u7ed9\u51fa\u4e86\u4f4d\u7f6e\u800c\u5219\u7ed9\u51fa\u4e86\u692d\u7403\u5728\u5404\u65b9\u5411\u4e0a\u7684\u8f74\u957f\u3002 \\textbf{\u591a\u4e2a\u692d\u7403\u533a\u57df}\u3002\u7531\u591a\u4e2aGauss\u5206\u5e03\u7684\u53e0\u52a0\uff0c\u5373\\textbf{\u6df7\u5408Gauss\u5206\u5e03}\u63cf\u8ff0\u3002\u6982\u7387\u5bc6\u5ea6\u5206\u5e03\u4e3a (5)   \u5206\u5e03\u63cf\u8ff0\u4e86\u6837\u672c\u7a7a\u95f4\u91cc\u591a\u4e2a\u7b80\u5355\u692d\u7403\u533a\u57df\uff0c\u6bcf\u4e2a\u533a\u57df\u53ef\u4ee5\u4ee3\u8868\u4e00\u4e2a\u6570\u636e\u7c7b\u3002\u901a\u5e38\u5728\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u628a\u89c2\u6d4b\u4e0d\u5230\u7684\u9690\u542b\u53d8\u91cf\uff0c\u7c7b\u522b\u6807\u7b7e\u52a0\u5165\u5230\u7edf\u8ba1\u91cf\u96c6\u5408\u4e2d\uff0c\u4e8e\u662f\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u6210\u4e3a (6)   \\textbf{\u51fd\u6570\u56fe\u5f62}\u3002\u51fd\u6570\u5728\u6837\u672c\u7a7a\u95f4\u91cc\u7684\u56fe\u5f62\u53ef\u4ee5\u7531\u6982\u7387\u5206\u5e03 (7)   \u523b\u753b\u3002\u7ed9\u5b9a\u4efb\u610f,\u90fd\u6ee1\u8db3\u4e00\u4e2a\u4ee5\u4e3a\u4e2d\u5fc3\u7684Gauss\u5206\u5e03\uff0c\u56e0\u800c\u8fd9\u91cc\u523b\u753b\u4e86\u4e00\u4e2a\u5bf9\u8fd9\u4e00\u80cc\u540e\u672c\u8d28\u7684\u6570\u91cf\u5173\u7cfb\u6240\u505a\u7684\u6d4b\u91cf\uff0c\u800c\u8be5\u6d4b\u91cf\u662f\u5e26\u6709Gauss\u566a\u58f0\u7684\u3002 \\subsection{\u6700\u5c0f\u4e8c\u4e58\u6cd5} \u8003\u8651\u51fd\u6570\u65cf,\u6211\u4eec\u901a\u8fc7\u4e00\u7cfb\u5217\u6d4b\u91cf\u6570\u636e\u53bb\u4f30\u8ba1\u80cc\u540e\u771f\u5b9e\u7684\u51fd\u6570\u5173\u7cfb. \u5047\u8bbe\u89c2\u6d4b\u566a\u58f0\u4e3aGauss\u5206\u5e03\uff0c\u4e8e\u662f\u89c2\u6d4b\u7684\u6982\u7387\u5206\u5e03\u4e3a (8)   \u6784\u9020\u5bf9\u6570\u4f3c\u7136\u51fd\u6570 (9)   \u4e8e\u662f\uff0c\u7ed9\u51fa (10)   \u5176\u4e2d. \u4e8e\u662f\u6211\u4eec\u5f97\u5230\u4e86\u6700\u5c0f\u4e8c\u4e58\u6cd5\u3002 \\subsection{EM\u7b97\u6cd5} \u5bf9\u4e8e\u4e00\u4e2a\u8054\u5408\u5206\u5e03\uff0c\u5b9e\u9645\u5e94\u7528\u4e2d\u5e76\u975e\u6bcf\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u90fd\u53ef\u4ee5\u88ab\u89c2\u6d4b\u5230\u3002\u6bd4\u5982\u4e0a\u8ff0\u6df7\u5408Gauss\u5206\u5e03\u4e2d\uff0c\u901a\u5e38\u7c7b\u522b\u6807\u7b7e\u662f\u65e0\u6cd5\u89c2\u6d4b\u5230\u7684\u3002\u8fd9\u79cd\u4e0d\u5b8c\u5907\u89c2\u6d4b\u60c5\u51b5\u4e0b\u76f4\u63a5\u4f7f\u7528\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4f1a\u51fa\u73b0\u8ba1\u7b97\u91cf\u8fc7\u5927\uff0c\u5b9e\u9645\u4e0d\u53ef\u89e3\u7684\u95ee\u9898\u3002\u4e8e\u662f\u5bf9\u4e8e\u8fd9\u7c7b\u4f30\u8ba1\u95ee\u9898\uff0c\u6211\u4eec\u4f7f\u7528\\textbf{EM\u7b97\u6cd5}\u6765\u6c42\u89e3\u3002 \u8003\u8651\u6df7\u5408Gauss\u5206\u5e03 (11)   \u8bbe\u4e3a\u89c2\u6d4b\u6837\u672c,\u4e3a\u89c2\u6d4b\u6837\u672c\u5bf9\u5e94\u7684\u9690\u53d8\u91cf\u3002\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u4e3a (12) 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