正規化された分配凾数(2)

正規化された分配凾数(2)

$$\begin{array}{rcl}{Z}_n(\beta )& =& {\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^{n}q(X_i;\theta {)}^{\beta }\mathrm{d}\theta \\ & =& {\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^n{\{q(X_i;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\}}^{\beta }\mathrm{d}\theta \cdots q(X_i;\theta {)}^{\beta }={\{q(X_i;{\theta }_0)\exp (-f(X_i,{\theta }_0,\theta ))\}}^{\beta }\\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^n\exp {(-f(X_i,{\theta }_0,\theta ))}^{\beta }\mathrm{d}\theta \\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\prod _{i=1}^n\exp (-\beta f(X_i,{\theta }_0,\theta ))\mathrm{d}\theta \\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-\beta \sum _{i=1}^{n}f(X_i,{\theta }_0,\theta ))\mathrm{d}\theta \cdots \prod _{i=1}^n\exp \left(A_i\right)=\exp \left(\sum _{i=1}^n{A}_i\right)\\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-\beta n{K}_n)\mathrm{d}\theta \cdots \sum _{i=1}^{n}f(X_i,{\theta }_0,\theta )=n{K}_n(\theta )\\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }){\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n)\mathrm{d}\theta \\ & =& (\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta })Z_n^{(0)}(\beta )\cdots Z_n^{(0)}(\beta )={\int }_{\mathrm{\Theta }}\pi (\theta )\exp (-n\beta K_n(\theta ))\mathrm{d}\theta :\mathrm{正}\mathrm{規}\mathrm{化}\mathrm{さ}\mathrm{れ}\mathrm{た}\mathrm{分}\mathrm{配}\mathrm{凾}\mathrm{数}\\ & =& Z_n^{(0)}(\beta )\prod _{i=1}^{n}q(X_i;{\theta }_0{)}^{\beta }\end{array}$$