正規化された分配凾数(2)
正規化された分配凾数(2)
$$ \begin{eqnarray} Z_n(\beta) &=&\int_\Theta \pi(\theta)\prod_{i=1}^n q(X_i;\theta)^\beta \mathrm{d}\theta\\ &=&\int_\Theta \pi(\theta)\prod_{i=1}^n \left\{q(X_i;\theta_0)\exp\left(-f(X_i,\theta_0,\theta)\right)\right\}^\beta \mathrm{d}\theta \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_1.html}{q(X_i;\theta)^\beta=\left\{q(X_i;\theta_0)\exp\left(-f(X_i,\theta_0,\theta)\right)\right\}^\beta}\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \prod_{i=1}^n \exp\left(-f(X_i,\theta_0,\theta)\right)^\beta \mathrm{d}\theta\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \prod_{i=1}^n \exp\left(-\beta f(X_i,\theta_0,\theta)\right) \mathrm{d}\theta\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \exp\left(-\beta \sum_{i=1}^n f(X_i,\theta_0,\theta)\right) \mathrm{d}\theta \;\cdots\;\prod_{i=1}^n \exp\left(A_i\right) = \exp \left(\sum_{i=1}^n A_i\right)\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \exp\left(-\beta\;nK_n\right) \mathrm{d}\theta \;\cdots\;\sum_{i=1}^n f(X_i,\theta_0,\theta)=nK_n(\theta)\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) \int_\Theta \pi(\theta) \exp\left(-n\beta K_n\right) \mathrm{d}\theta\\ &=&\left(\prod_{i=1}^n q(X_i;\theta_0)^\beta\right) Z_n^{(0)}(\beta) \;\cdots\;Z_n^{(0)}(\beta)=\int_\Theta \pi(\theta)\exp(-n\beta K_n(\theta))\mathrm{d}\theta:正規化された分配凾数\\ &=& Z_n^{(0)}(\beta)\prod_{i=1}^n q(X_i;\theta_0)^\beta\\ \end{eqnarray} $$正規化された分配凾数
正規化された分配凾数
$$ \begin{eqnarray} \prod^n_{i=1}q(X_i;\theta)^{\beta} &=&\exp\left(\log\left(\displaystyle\prod^n_{i=1}q(X_i;\theta)^{\beta}\right)\right) \;\cdots\;A=\exp\left(\log\left(A\right)\right)\\ &=&\exp\left(\sum^n_{i=1}\log\left(q(X_i;\theta)^{\beta}\right)\right) \;\cdots\;\log\left(\prod_{i=1}^n A_i\right) = \sum_{i=1}^n \log\left(A_i\right)\\ &=&\exp\left(\beta\sum^n_{i=1}\log\left(q(X_i;\theta)\right)\right)\\ &=&\exp\left(-n\beta \left(-\frac{1}{n}\sum^n_{i=1}\log\left(q\left(X_i;\theta\right)\right)\right)\right) \;\cdots\;(-n)(-\frac{1}{n})=1\\ &=&\exp\left(-n\beta L_n(\theta)\right) \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_1.html}{L_n(\theta)=-\frac{1}{n}\displaystyle\sum^n_{i=1}\log\left(q(X_i;\theta)\right)}\\ \end{eqnarray} $$$$ \begin{eqnarray} Z_n(\beta) &=&\int_\Theta \pi(\theta) \prod^n_{i=1}q(X_i;\theta)^{\beta} \mathrm{d}\theta\\ &=&\int_\Theta \pi(\theta) \exp\left(-n\beta L_n(\theta)\right) \mathrm{d}\theta \;\cdots\;\prod^n_{i=1}q(X_i;\theta)^{\beta}=\exp\left(-n\beta L_n(\theta)\right)\\ &=&\int_\Theta \pi(\theta) \exp\left(-n\beta \left( L_n(\theta_0) + K_n(\theta) \right)\right) \mathrm{d}\theta \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_1.html}{L_n(\theta) = L_n(\theta_0) + K_n(\theta)}\\ &=&\int_\Theta \pi(\theta) \exp(-n\beta L_n(\theta_0) -n\beta K_n(\theta)) \mathrm{d}\theta\\ &=&\int_\Theta \pi(\theta) \exp(-n\beta L_n(\theta_0)) \exp(-n\beta K_n(\theta)) \mathrm{d}\theta\\ &=&\exp(-n\beta L_n(\theta_0))\int_\Theta \pi(\theta)\exp(-n\beta K_n(\theta)) \mathrm{d}\theta\\ &=&\exp(-n\beta L_n(\theta_0))Z_n^{(0)}(\beta) \;\cdots\;Z_n^{(0)}(\beta)=\int_\Theta \pi(\theta)\exp(-n\beta K_n(\theta)) \mathrm{d}\theta:正規化された分配凾数\\ &=& Z_n^{(0)}(\beta)\exp(-n\beta L_n(\theta_0))\\ \end{eqnarray} $$
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