李航-统计学习方法-习题-第九章

9.2 证明引理 9.2.
引理 9.2 若${\tilde{P}}_{\theta }(Z)=P(Z|Y,\theta )$，则
$$F(\tilde{P},\theta )=logP(Y|\theta )$$.
证明：
$$\begin{array}{rl}F(\tilde{P},\theta )& =E_{\tilde{P}}[logP(Y,Z|\theta )]+H(\tilde{P})\\ & =E_{\tilde{P}}[logP(Y,Z|\theta )]-E_{\tilde{P}}log\tilde{P}(Z)\\ & =\sum _{Z}logP(Y,Z|\theta ){\tilde{P}}_{\theta }(Z)-\sum _{Z}log\tilde{P}(Z)\tilde{P}(Z)\\ & =\sum _{Z}logP(Y,Z|\theta )P(Z|Y,\theta )-\sum _{Z}logP(Z|Y,\theta )P(Z|Y,\theta )\\ & =\sum _{Z}P(Z|Y,\theta )(logP(Y,Z|\theta )-logP(Z|Y,\theta ))\\ & =\sum _{Z}P(Z|Y,\theta )log\frac{P(Y,Z|\theta )}{P(Z|Y,\theta )}\\ & =\sum _{Z}P(Z|Y,\theta )logP(Y|\theta )\\ & =logP(Y|\theta )\sum _{Z}P(Z|Y,\theta )\\ & =logP(Y|\theta )\cdot 1\\ & =logP(Y|\theta )\end{array}$$
证毕.