Exercise 7.2 Solution Example - Hoff, A First Course in Bayesian Statistical Methods
標準ベイズ統計学 演習問題 7.2 解答例

\begin{align*} l(\boldsymbol{\theta}, \Psi | \boldsymbol{Y}) &= \sum_{i=1}^n \log p(\boldsymbol{y}_i | \boldsymbol{\theta}, \Psi) \\ &= \sum_{i=1}^n \log \left\{ (2 \pi)^{-p/2} |\Psi|^{1/2} \exp \left( -\frac{1}{2} (\boldsymbol{y}_i - \boldsymbol{\theta})^T \Psi (\boldsymbol{y}_i - \boldsymbol{\theta}) \right) \right\} \\ &= \sum_{i=1}^n \left\{ -\frac{p}{2} \log (2 \pi) + \frac{1}{2} \log |\Psi| -\frac{1}{2} (\boldsymbol{y}_i - \boldsymbol{\theta})^T \Psi (\boldsymbol{y}_i - \boldsymbol{\theta}) \right\} \\ &= -\frac{np}{2} \log (2 \pi) + \frac{n}{2} \log |\Psi| -\frac{1}{2} \sum_{i=1}^n (\boldsymbol{y}_i - \boldsymbol{\theta})^T \Psi (\boldsymbol{y}_i - \boldsymbol{\theta}) \\ &= -\frac{np}{2} \log (2 \pi) + \frac{n}{2} \log |\Psi| -\frac{1}{2} \sum_{i=1}^n \left\{ (\boldsymbol{y}_i - \bar{\boldsymbol{y} } + \bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\boldsymbol{y}_i - \bar{\boldsymbol{y} } + \bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \\ &= -\frac{np}{2} \log (2 \pi) + \frac{n}{2} \log |\Psi| -\frac{1}{2} \sum_{i=1}^n \left\{ ( (\boldsymbol{y}_i - \bar{\boldsymbol{y} })^T + ( \bar{\boldsymbol{y} } - \boldsymbol{\theta})^T) \Psi ((\boldsymbol{y}_i - \bar{\boldsymbol{y} }) + (\bar{\boldsymbol{y} } - \boldsymbol{\theta})) \right\} \\ &= -\frac{np}{2} \log (2 \pi) + \frac{n}{2} \log |\Psi| \\ &\qquad -\frac{1}{2} \sum_{i=1}^n \left\{ (\boldsymbol{y}_i - \bar{\boldsymbol{y} })^T \Psi (\boldsymbol{y}_i - \bar{\boldsymbol{y} }) + (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) + 2 (\boldsymbol{y}_i - \bar{\boldsymbol{y} })^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \\ &= -\frac{np}{2} \log (2 \pi) + \frac{n}{2} \log |\Psi| -\frac{1}{2} \sum_{i=1}^n \left\{ (\boldsymbol{y}_i - \bar{\boldsymbol{y} })^T \Psi (\boldsymbol{y}_i - \bar{\boldsymbol{y} }) + (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \\ &= -\frac{np}{2} \log (2 \pi) + \frac{n}{2} \log |\Psi| -\frac{n}{2} \text{tr}(S \Psi) -\frac{n}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}), \\ &\qquad \qquad\text{where} \quad S = \frac{1}{n} \sum_{i=1}^n (\boldsymbol{y}_i - \bar{\boldsymbol{y} }) (\boldsymbol{y}_i - \bar{\boldsymbol{y} })^T \\ \end{align*}

より、

\begin{align*} \log p_U(\boldsymbol{\theta}, \Psi) &= \frac{ l(\boldsymbol{\theta}, \Psi | \boldsymbol{Y}) }{n} + c \\ &= -\frac{p}{2} \log (2 \pi) + \frac{1}{2} \log |\Psi| -\frac{1}{2} \text{tr}(S \Psi) -\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) + c \\ \end{align*}

となる。よって

\begin{align*} p_U(\boldsymbol{\theta}, \Psi) &= \exp \left\{ -\frac{p}{2} \log (2 \pi) + \frac{1}{2} \log |\Psi| -\frac{1}{2} \text{tr}(S \Psi) -\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) + c \right\} \\ &= (2 \pi)^{- \frac{p}{2} } |\Psi|^{ \frac{1}{2} } \exp \left\{ -\frac{1}{2} \text{tr}(S \Psi) \right\} \exp \left\{-\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \exp(c)\\ &\propto (2 \pi)^{- \frac{p}{2} } |\Psi^{-1}|^{- \frac{1}{2} } \exp \left\{-\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \times |\Psi|^{ \frac{p+1-p-1}{2} } \exp \left\{ -\frac{1}{2} \text{tr}(S \Psi) \right\} \\ &\propto (2 \pi)^{- \frac{p}{2} } |\Psi^{-1}|^{- \frac{1}{2} } \exp \left\{-\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Psi (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \times |\Psi|^{ \frac{p+1-p - 1}{2} } \exp \left\{ -\frac{1}{2} \text{tr}(S \Psi) \right\} \\ &\propto \text{dmultivariate-normal}(\boldsymbol{\theta} | \bar{ \boldsymbol{y} } , \Psi^{-1}) \times \text{dWishart}(\Psi | p+1, S^{-1}) \\ \end{align*}

が成立。

\begin{align*} p_U(\boldsymbol{\theta}, \boldsymbol{\Sigma} | \boldsymbol{y}_1, \ldots, \boldsymbol{y}_n) &\propto p_U(\boldsymbol{\theta} | \boldsymbol{\Sigma}) p_U(\boldsymbol{\Sigma}) p(\boldsymbol{y}_1, \ldots, \boldsymbol{y}_n | \boldsymbol{\theta}, \boldsymbol{\Sigma}) \\ &\propto |\boldsymbol{\Sigma}|^{- \frac{1}{2} } \exp \left\{-\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \boldsymbol{\Sigma}^{-1} (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \times |\boldsymbol{\Sigma}^{-1}|^{ \frac{p+1-p - 1}{2} } \exp \left\{ -\frac{1}{2} \text{tr}(S \boldsymbol{\Sigma}^{-1}) \right\} \\ & \qquad \times |\Sigma|^{ -\frac{n}{2} } \exp \left\{ -\frac{1}{2} \sum_{i=1}^n (\boldsymbol{y}_i - \boldsymbol{\theta})^T \Sigma^{-1} (\boldsymbol{y}_i - \boldsymbol{\theta}) \right\} \\ &\propto |\boldsymbol{\Sigma}|^{- \frac{1}{2} } \exp \left\{-\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \boldsymbol{\Sigma}^{-1} (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \times \exp \left\{ -\frac{1}{2} \text{tr}(S \boldsymbol{\Sigma}^{-1}) \right\} \\ & \qquad \times |\Sigma|^{ -\frac{n}{2} } \exp \left\{ -\frac{1}{2} \sum_{i=1}^n (\boldsymbol{y}_i - \bar{y} + \bar{y} - \boldsymbol{\theta})^T \Sigma^{-1} (\boldsymbol{y}_i - \bar{y} + \bar{y} - \boldsymbol{\theta}) \right\} \\ &\propto |\boldsymbol{\Sigma}|^{- \frac{1}{2} } \exp \left\{-\frac{1}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \boldsymbol{\Sigma}^{-1} (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \times \exp \left\{ -\frac{1}{2} \text{tr}(S \boldsymbol{\Sigma}^{-1}) \right\} \\ & \qquad \times |\Sigma|^{ -\frac{n}{2} } \exp \left\{ - \frac{n}{2} \text{tr}(S \boldsymbol{\Sigma}^{-1}) \right\} \exp \left\{-\frac{n}{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \Sigma^{-1} (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \\ & \propto |\boldsymbol{\Sigma}|^{- \frac{n+1}{2} } \exp \left\{-\frac{ n+1 }{2} (\bar{\boldsymbol{y} } - \boldsymbol{\theta})^T \boldsymbol{\Sigma}^{-1} (\bar{\boldsymbol{y} } - \boldsymbol{\theta}) \right\} \times \exp \left\{ -\frac{n+1}{2} \text{tr}(S \boldsymbol{\Sigma}^{-1}) \right\} \\ & \propto |\boldsymbol{\Sigma}|^{- \frac{n+1}{2} } \exp \left\{-\frac{1}{2} (\boldsymbol{\theta} - \bar{\boldsymbol{y} })^T (n+1) \boldsymbol{\Sigma}^{-1} (\boldsymbol{\theta} - \bar{\boldsymbol{y} }) \right\} \times \exp \left\{ -\frac{1}{2} \text{tr}((n+1) S \boldsymbol{\Sigma}^{-1}) \right\} \\ & \propto | \frac{1}{n+1} \boldsymbol{\Sigma}|^{- \frac{1}{2} } \exp \left\{-\frac{1}{2} (\boldsymbol{\theta} - \bar{\boldsymbol{y} })^T ( \frac{1}{n+1} \boldsymbol{\Sigma})^{-1} (\boldsymbol{\theta} - \bar{\boldsymbol{y} }) \right\} \\ &\qquad \times |\Sigma|^{- \frac{n-p-1+p+1}{2} }\exp \left\{ -\frac{1}{2} \text{tr}((n+1) S \boldsymbol{\Sigma}^{-1}) \right\} \\ & \propto \text{dmultivariate-normal}(\boldsymbol{\theta} | \bar{\boldsymbol{y} }, \frac{1}{n+1} \boldsymbol{\Sigma}) \times \text{dinverse-Wishart}(\boldsymbol{\Sigma} | n-p-1, \frac{1}{n+1} S^{-1}) \\ \end{align*}

上式より、 \(p_U(\boldsymbol{\theta}, \boldsymbol{\Sigma} | \boldsymbol{y}_1, \ldots, \boldsymbol{y}_n)\) は、データと事前分布の情報が反映された分布であることがわかるので、事後分布とみなせる。

(From the expression above, we can see that \(p_U(\boldsymbol{\theta}, \boldsymbol{\Sigma} | \boldsymbol{y}_1, \ldots, \boldsymbol{y}_n)\) is a distribution that reflects information from both the data and the prior distribution. Therefore, it can be regarded as the posterior distribution.)