正規化された分配凾数(2)

正規化された分配凾数(2)

$$ \begin{eqnarray} Z_n(\beta) &=&\int_\Theta \pi(\theta)\prod_{i=1}^n q(X_i;\theta)^\beta \mathrm{d}\theta\\ &=&\int_\Theta \pi(\theta)\prod_{i=1}^n \left\{q(X_i;\theta_0)\exp\left(-f(X_i,\theta_0,\theta)\right)\right\}^\beta \mathrm{d}\theta \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_1.html}{q(X_i;\theta)^\beta=\left\{q(X_i;\theta_0)\exp\left(-f(X_i,\theta_0,\theta)\right)\right\}^\beta}\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \prod_{i=1}^n \exp\left(-f(X_i,\theta_0,\theta)\right)^\beta \mathrm{d}\theta\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \prod_{i=1}^n \exp\left(-\beta f(X_i,\theta_0,\theta)\right) \mathrm{d}\theta\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \exp\left(-\beta \sum_{i=1}^n f(X_i,\theta_0,\theta)\right) \mathrm{d}\theta \;\cdots\;\prod_{i=1}^n \exp\left(A_i\right) = \exp \left(\sum_{i=1}^n A_i\right)\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \exp\left(-\beta\;nK_n\right) \mathrm{d}\theta \;\cdots\;\sum_{i=1}^n f(X_i,\theta_0,\theta)=nK_n(\theta)\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \exp\left(-n\beta K_n\right) \mathrm{d}\theta\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) Z_n^{(0)}(\beta) \;\cdots\;Z_n^{(0)}(\beta)=\int_\Theta \pi(\theta)\exp(-n\beta K_n(\theta))\mathrm{d}\theta:正規化された分配凾数\\ &=& Z_n^{(0)}(\beta)\prod_{i=1}^n q(X_i;\theta_0)^\beta\\ \end{eqnarray} $$