尤度凾数の期待値の平均情報量(連続)

尤度凾数の期待値の平均情報量(連続)

$$\begin{array}{rcl}{G}_n& =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;\theta )]\right)]\cdots \mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}\\ & =& -{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots q(X;\theta )=q(X;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\\ & =& -{\mathrm{E}}_X[\log (q(X;{\theta }_0){\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))])]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& -{\mathrm{E}}_X[\log (q(X;{\theta }_0))+\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots \log \left(AB\right)=\log \left(A\right)+\log \left(B\right)\\ & =& -{\mathrm{E}}_X[\log (q(X;{\theta }_0))]-{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots \mathrm{E}[X+Y]=\mathrm{E}\left[X\right]+\mathrm{E}\left[Y\right]\\ & =& L({\theta }_0)-{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]\cdots -{\mathrm{E}}_X[\log (q(X;{\theta }_0))]=L({\theta }_0)\\ & =& L({\theta }_0)+G_n^{(0)}\cdots G_n^{(0)}=-{\mathrm{E}}_X[\log \left({\mathrm{E}}_{\mathrm{\Theta }}[\exp (-f(X_i,{\theta }_0,\theta ))]\right)]:\mathrm{正}\mathrm{規}\mathrm{化}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{尤}\mathrm{度}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{期}\mathrm{待}\mathrm{値}\mathrm{の}\mathrm{平}\mathrm{均}\mathrm{情}\mathrm{報}\mathrm{量}\end{array}$$

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