尤度凾数の期待値の平均情報量(離散/標本からの)

尤度凾数の期待値の平均情報量(離散/標本からの)

$$\begin{array}{rcl}{T}_n& =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;\theta )]\right)\cdots \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))]\right)\cdots q(X;\theta )=q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\\ & =& -\frac{1}{n}\sum _{i=1}^n\log (q(X;{\theta }_0){\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))])\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& -\frac{1}{n}\sum _{i=1}^n\{\log (q(X;{\theta }_0))+\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\}\cdots \log \left(AB\right)=\log \left(A\right)+\log \left(B\right)\\ & =& -\frac{1}{n}\{\sum _{i=1}^n\log (q(X;{\theta }_0))+\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\}\cdots \sum (A+B)=\sum A+\sum B\\ & =& -\frac{1}{n}\sum _{i=1}^n\log (q(X;{\theta }_0))-\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\\ & =& L_n({\theta }_0)-\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\cdots -\frac{1}{n}\sum _{i=1}^n\log (q(X;{\theta }_0))=L_n({\theta }_0)\\ & =& L_n({\theta }_0)+T_n^{(0)}\cdots -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)=T_n^{(0)}\end{array}$$

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$$\begin{array}{rcl}{T}_n^{(0)}& =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-\log \frac{q(X_i;{\theta }_0)}{q(X_i;\theta )})]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (\log \frac{q(X_i;\theta )}{q(X_i;{\theta }_0)})]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left({\mathrm{E}}_{\mathrm{\Theta }}\left[\frac{q(X_i;\theta )}{q(X_i;{\theta }_0)}\right]\right)\\ & =& -\frac{1}{n}\sum _{i=1}^n\log \left(\frac{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}{q(X_i;{\theta }_0)}\right)\\ & =& \frac{1}{n}\sum _{i=1}^n-\log \left(\frac{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}{q(X_i;{\theta }_0)}\right)\\ & =& \frac{1}{n}\sum _{i=1}^n\log \left(\frac{q(X_i;{\theta }_0)}{{\mathrm{E}}_{\mathrm{\Theta }}[q(X_i;\theta )]}\right)\end{array}$$