{"id":897,"date":"2018-12-08T14:24:42","date_gmt":"2018-12-08T06:24:42","guid":{"rendered":"http:\/\/planckscale.info\/?p=897"},"modified":"2018-12-08T15:06:43","modified_gmt":"2018-12-08T07:06:43","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%ba%8c-em%e7%ae%97%e6%b3%95%e5%ae%9e%e4%be%8b%ef%bc%9a-%e5%bd%a2%e7%8a%b6%e7%ba%a6%e6%9d%9f","status":"publish","type":"post","link":"https:\/\/mindspectrum.xyz\/en\/2018\/12\/08\/%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%ba%8c-em%e7%ae%97%e6%b3%95%e5%ae%9e%e4%be%8b%ef%bc%9a-%e5%bd%a2%e7%8a%b6%e7%ba%a6%e6%9d%9f\/","title":{"rendered":"\u7edf\u8ba1\u7b97\u6cd5\u7684\u7406\u8bba\u4e0e\u5b9e\u4f8b\u4e4b\u4e8c EM\u7b97\u6cd5\u5b9e\u4f8b\uff1a \u5f62\u72b6\u7ea6\u675f\u7684\u975e\u53c2\u6570\u6df7\u5408\u9ad8\u65af\u6a21\u578b"},"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<\/a>)\u3001\u4f5c\u8005\u4fe1\u606f\u548c\u672c\u58f0\u660e\uff0c\u5426\u5219\u5c06\u8ffd\u7a76\u6cd5\u5f8b\u8d23\u4efb\u3002<\/strong><\/h5>\n

\u6700\u8fd1\u6bd4\u8f83\u65e0\u804a\uff0c\u5e72\u8106\u628a\u8fd9\u4e2a\u7cfb\u5217\u8865\u5b8c\u3002\u672c\u6587\u7684\u5185\u5bb9\u662f\u6211\u8868\u59b9\u4e4b\u524d\u5728\u829d\u5927\u8bfb\u4e66\u65f6\u7684\u4e00\u4e2a\u5c0f\u9898\u76ee\uff0c\u4f5c\u4e3aEM\u7b97\u6cd5\u7684\u4e00\u4e2a\u6bd4\u8f83\u590d\u6742\u7684\u5b9e\u4f8b\u6765\u6f14\u793a\u662f\u4e0d\u9519\u7684\u3002\u7531\u4e8e\u6211\u6ca1\u627e\u5230\u5f15\u6587\uff0c\u5c31\u4e0d\u5f15\u7528\u4e86\u3002\u6211\u4e0d\u662f\u505a\u7edf\u8ba1\u7684\uff0c\u6709\u9519\u8bef\u6b22\u8fce\u7559\u8a00\u3002<\/p>\n

\\subsubsection{\u89e3\u6790\u63a8\u5bfc}
\n\u8fd9\u662f\u4e00\u4e2a\u66f4\u4e3a\u590d\u6742\u7684\u4f30\u8ba1\u95ee\u9898\u5b9e\u4f8b\u3002\u8981\u4f30\u8ba1\u7684\u5206\u5e03\u662f\u6837\u672c\u7a7a\u95f4\u91cc\u4e00\u7ec4\u51fd\u6570\u56fe\u5f62\u5bf9\u5e94\u7684\u5206\u5e03\u3002\u5177\u4f53\u8bf4\u6765\uff0c\u6211\u4eec\u6709\u968f\u673a\u53d8\u91cf\"\{\boldsymbol{X},Y,Z\}\",\u5176\u4e2d\"\boldsymbol{X}\in \mathbf{R}^p\",\"Y\in \mathbf{R}\",\"Z \in \{1,...K\}\",\u5176\u4e2d\"Z\"\u662f\u6df7\u5408\u9ad8\u65af\u6a21\u578b\u4e2d\u7684\u7c7b\u522b\u6807\u7b7e\u53d8\u91cf,\u5bf9\u4e8e\u4e0d\u540c\u7684\"Z\"\u503c,\"Y\"\u548c\"\boldsymbol{X}\"\u5b58\u5728\u4e00\u4e2a\u4e0d\u540c\u7684\u6f5c\u5728\u51fd\u6570\u5173\u7cfb\"m_Z(x)\",\u5176\u4e2d\"m_Z(x)\"\u4e3a\u51f8\u51fd\u6570\u3002\u6982\u7387\u6a21\u578b\u8868\u8ff0\u4e3a<\/p>\n

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

\u73b0\u5728\u5047\u5b9a\"Z\"\u4e0d\u53ef\u89c2\u6d4b\uff0c\u6211\u4eec\u4ece\u4e00\u7cfb\u5217\u89c2\u6d4b\u503c\"\{\boldsymbol{x}_i,y_i\}\"\u4e2d\u4f30\u8ba1\u51fa\u6a21\u578b\u7684\u53c2\u6570\"\boldsymbol{\theta}\equiv \{\boldsymbol{\pi},\boldsymbol{m(x)},\boldsymbol{\Sigma}=\{\sigma_1,...\sigma_{K}\}\}\"\u7684\u771f\u5b9e\u503c\"\boldsymbol{\theta}_*\"\u3002<\/p>\n

\u63a5\u4e0b\u6765\u6211\u4eec\u7528EM\u7b97\u6cd5\u6765\u5904\u7406\u8fd9\u4e00\u95ee\u9898\u3002\u8bbe\"\mathcal{X}\equiv\{\boldsymbol{x}_{1},...\boldsymbol{x}_N\},\mathcal{Y}\equiv \{y_1,...y_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\u6211\u4eec\u9996\u5148\u8ba1\u7b97\"p(\mathcal{Z}\vert \mathcal{X},\mathcal{Y},\boldsymbol{\theta}^{(t)})\"<\/p>\n

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

\u7c7b\u4f3c\u4e8e\u8fb9\u7f18\u5206\u5e03\u548c\u6761\u4ef6\u5206\u5e03\u7684\u56e0\u5b50\u5206\u89e3\u6027\u8d28\uff0c\u53ef\u4ee5\u8bc1\u660e\u5bf9\u4efb\u610f\u6ee1\u8db3\"q(\mathcal{Z})=\prod_i q_i(z_i)\"\u7684\u6982\u7387\u5206\u5e03\"q(\mathcal{Z})\",\u6709<\/p>\n

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

\u8fd9\u610f\u5473\u7740\"Q(\boldsymbol{\theta}\vert\boldsymbol{\theta}^{(t)})\"\u53ef\u4ee5\u5206\u89e3\u6210\u5728\u6bcf\u4e2a\u6837\u672c\u4e0a\u5404\u81ea\u5747\u503c\u4e4b\u603b\u548c\u7684\u5f62\u5f0f<\/p>\n

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

\u6211\u4eec\u628a\u8be5\u6027\u8d28\u7684\u8bc1\u660e\u7559\u5230\u540e\u9762\u3002<\/p>\n

\u4e8e\u662f<\/p>\n

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

\u9700\u8981\u6ce8\u610f\u7684\u662f\"\boldsymbol{m}(\boldsymbol{x})=\{\boldsymbol{m}_1(\boldsymbol{x}),...\boldsymbol{m}_{K}(\boldsymbol{x})\}\"\u5305\u542b\u4e86\u65e0\u7a77\u4e2a\u53c2\u6570\u9700\u8981\u4f18\u5316\uff0c\u4e3a\u4e86\u7ea6\u5316\u8fd9\u4e00\u95ee\u9898\uff0c\u6211\u4eec\u5c06\u5176\u79bb\u6563\u5316\uff0c\u53d6\"m_{ji}=m(\boldsymbol{x}_i)\"\u4e3a\u4f18\u5316\u53c2\u6570\uff0c\u4ecd\u7136\u7b80\u8bb0\u4e3a\"\boldsymbol{m}\".\u8fd9\u6837\uff0c\"\boldsymbol{m}(\boldsymbol{x})\"\u88ab\u7ea6\u5316\u5230\u7528\"K\times N\"\u4e2a\u53c2\u6570\u6765\u8868\u793a\u3002<\/p>\n

\u63a5\u4e0b\u6765\u6211\u4eec\u9700\u8981\u6c42\u89e3\u7684\u6781\u5927\u5316\u95ee\u9898\u4e3a<\/p>\n

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

\u8fd9\u91cc\u6211\u4eec\u4e3a\"m(x)\"\u52a0\u5165\u4e86\u51f8\u51fd\u6570\u7684\u5f62\u72b6\u7ea6\u675f\u3002\u7ea6\u675f\u7684\u51e0\u4f55\u610f\u4e49\u5f88\u660e\u786e\uff0c\u5373\\textbf{\u51fd\u6570\u603b\u662f\u4f4d\u4e8e\u6bcf\u4e00\u70b9\u7684\u5207\u7ebf\u4e4b\u4e0a\u65b9}\u3002<\/p>\n

\u7531\u4e8e\"Q(\boldsymbol{\theta}\vert\boldsymbol{\theta}^{(t)})\"\u4e2d\"\boldsymbol{\pi}\"\u7684\u6781\u5927\u5316\u548c\u5176\u4ed6\u53c2\u6570\u76f8\u4e92\u72ec\u7acb\uff0c\u56e0\u800c\u53ef\u5c06\u95ee\u9898\u89e3\u8026\u3002\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898\uff0c\u5176\u4e2d\"\boldsymbol{\pi}\"\u53c2\u6570\u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a<\/p>\n

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

\u53ef\u4ee5\u6c42\u51fa\u89e3\u6790\u89e3\uff0c\u5373<\/p>\n

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

\"\boldsymbol{m}\"\u53c2\u6570\u5bf9\u5e94\u7684\u5b50\u95ee\u9898<\/p>\n

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

\u662f\u4e2a\u4e8c\u6b21\u89c4\u5212\u95ee\u9898\u3002\u7531\u4e8e\"\sigma_j,m_{ji}\"\u53ea\u51fa\u73b0\u5728\u5177\u6709\u76f8\u540c\"j\"\u503c\u7684\u9879\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u6309\u4e0d\u540c\u7684\"j\"\u503c\u628a\u9879\u5206\u7ec4\uff0c\u6bcf\u4e00\u7ec4\u90fd\u662f\u72ec\u7acb\u53d8\u5316\u7684\uff0c\u56e0\u800c\u53ef\u4ee5\u4f5c\u4e3a\u4e00\u4e2a\u72ec\u7acb\u7684\u5b50\u95ee\u9898\u6765\u6700\u5927\u5316\u3002\u53ef\u4ee5\u53d1\u73b0\uff0c\\textbf{\u8fd9\u91cc\u7684Gauss\u6df7\u5408\u6a21\u578b\u4e2d\u6bcf\u4e2acomponent\u90fd\u5bf9\u5e94\u4e8e\u4e00\u4e2a\u5e26\u7ea6\u675f\u7684\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898}\u3002\u8fd9\u4e00\u6027\u8d28\u53ef\u4ee5\u7528\u6765\u964d\u4f4e\u8ba1\u7b97\u590d\u6742\u5ea6\uff0c\u63d0\u9ad8\u7b97\u6cd5\u6548\u7387\u3002<\/p>\n

\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\uff0c\u4e8c\u6b21\u89c4\u5212\u4e2d\u7684\u7ea6\u675f\u53ef\u4ee5\u5927\u5e45\u5ea6\u7684\u7ea6\u51cf\u3002\u7b2c\u4e00\u79cd\u60c5\u51b5\u662f\u5f53\"X\"\u4e3a\u5355\u53d8\u91cf\u65f6\u3002\u4e0d\u5931\u4e00\u822c\u6027\u7684\uff0c\u6211\u4eec\u5047\u8bbe\u89c2\u6d4b\u6837\u672c\u5df2\u7ecf\u6309\u7167\"X\"\u7684\u5927\u5c0f\u505a\u5347\u5e8f\u6392\u5217\uff0c\u6b64\u65f6\u53ef\u4ee5\u7528\u4e8c\u9636\u5bfc\u6570\u975e\u8d1f\u6765\u4f5c\u4e3a\u51f8\u6027\u7ea6\u675f\u3002\u4e8e\u662f\u7ea6\u675f\u53ef\u7ea6\u5316\u4e3a
\n<\/a><\/p>\n

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

\u8fd9\u91cc\"\beta_{ji}\"\u4ee3\u8868\u4e86\u7b2c\"j\"\u4e2acomponent\u5728\u7b2c\"x_i\"\u5904\u5bfc\u6570\u7684\u4f30\u8ba1\u503c\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e3a\u4e86\u5f97\u5230\u4e00\u70b9\u4e0a\u5bfc\u6570\uff08\u5207\u7ebf\uff09\u7684\u4f30\u8ba1\uff0c\u6211\u4eec\u9700\u8981\u4e24\u4e2a\u7ea6\u675f\uff0c\u7ea6\u675f\u5728\u5de6\u53f3\u4e24\u4fa7\u7684\u51fd\u6570\u503c\u90fd\u8981\u4f4d\u4e8e\u5207\u7ebf\u4e0a\u65b9\u3002<\/p>\n

\u53e6\u4e00\u79cd\u60c5\u51b5\u5219\u662f\u5f53\"X\"\u4e3a\u591a\u7ef4\u7684\u5411\u91cf\uff0c\u4e14\u6709\"m(\boldsymbol{X})=\sum_\alpha m^\alpha(X_\alpha)\"\u65f6\u3002\u6b64\u65f6\u6bcf\u4e2a\u5206\u91cf\"m^\alpha(X_\alpha)\"\u7684\u51f8\u6027\u610f\u5473\u7740\"m(\boldsymbol{X})\"\u4e5f\u662f\u51f8\u7684\uff0c\u56e0\u6b64\u7ea6\u675f\u53ef\u5199\u4e3a<\/p>\n

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

\u5373\u5bf9\u5411\u91cf\u51fd\u6570\u7684\u6bcf\u4e2a\u5206\u91cf\uff0c\u90fd\u6ee1\u8db3\\eqref{mConstraint1D}\u6240\u793a\u7684\u7ea6\u675f\u3002<\/p>\n

\u6700\u540e\u6211\u4eec\u6765\u8bc1\u660e\"E_{q(\mathcal{Z})}[\log L(\mathcal{X},\mathcal{Z})]\"\u7684\u5206\u89e3\u6027\u8d28\uff0c\u5176\u4e2d\"q(\mathcal{Z})\"\u4e3a\u4efb\u610f\u6ee1\u8db3\"q(\mathcal{Z})=\prod_i q_i(z_i)\"\u7684\u6982\u7387\u5206\u5e03<\/p>\n

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

\u53ef\u89c1\u5bf9\u6bcf\u4e2asample\uff0c\u90fd\u9700\u8981\u5728\u5206\u5e03\"q(\mathcal{Z})\"\u4e0a\u6c42\"\log p(\boldsymbol{x}_i,z_i)\"\u7684\u5747\u503c\u3002\u4f46\u7531\u4e8e\"\mathcal{Z}\"\u4e2d\u5305\u542b\u5927\u91cf\u65e0\u5173\u7684\"z_k\neq z_i\",\u5b83\u4eec\u90fd\u5e94\u8be5\u88ab\u52a0\u548c\u4e3a1,\u4ece\u800c\u53ea\u5269\u4e0b\"q_i(z_i)\":<\/p>\n

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

\\subsubsection{\u7b97\u6cd5\u5b9e\u73b0}
\n\u7b97\u6cd5\u5b9e\u73b0\u4e3b\u8981\u9700\u8981\u6ce8\u610f\u7684\u662f\u9632\u6b62\u6570\u503c\u4e0b\u6ea2\u5e26\u6765\u7684\u9664\u96f6\u95ee\u9898\u3002\u6280\u672f\u4e0a\uff0c\u907f\u514d\u6982\u7387\u8fde\u4e58\u79ef\uff0c\u907f\u514d\u6307\u6570\uff0c\u8ba9\u8ba1\u7b97\u7684\u4e2d\u95f4\u7ed3\u679c\u5c3d\u91cf\u5728\u5bf9\u6570\u57df\u4e2d\u8868\u793a\u662f\u4e00\u4e9b\u57fa\u672c\u6280\u5de7\u3002<\/p>\n

\u9996\u5148\uff0c\u5728\u8ba1\u7b97\"\gamma_{ij}\"\u65f6\uff0c\u53ef\u80fd\u4f1a\u51fa\u73b0\u67d0\u4e2a\u6837\u672c\u70b9\u8fdc\u79bb\u6240\u6709component\u7684\u60c5\u51b5\uff0c\u6b64\u65f6\u5206\u5b50\u5206\u6bcd\u90fd\u5bb9\u6613\u4e0b\u6ea2\u4e3a\u96f6\uff0c\u800c\u51fa\u73b0\"\frac{0}{0}\"\u7684\u60c5\u51b5\u3002\u6211\u4eec\u505a\u5982\u4e0b\u5904\u7406<\/p>\n

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

\u7b97\u6cd5\u4e2d\u9700\u8981\u8ba1\u7b97\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u7684\u4e0a\u5347\u7387\uff0c\u6765\u5224\u65ad\u7b97\u6cd5\u7684\u7ec8\u6b62\u6761\u4ef6\u662f\u5426\u6ee1\u8db3\u3002\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u4e2d\u6709\u5927\u91cf\u7684\u6982\u7387\u8fde\u4e58\u79ef\uff0c\u6211\u4eec\u4f5c\u5982\u4e0b\u5904\u7406\u6765\u6d88\u9664\u4e4b<\/p>\n

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

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