\[
\newcommand\l{\left}
\newcommand\r{\right}
\newcommand\cmt[1]{\class{Cmt}{\mbox{#1}}}
\newcommand\b[1]{\class{Bold}{\mathrm{#1}}}
\newcommand\bA{\b{A}}
\newcommand\bw{\b{w}}
\newcommand\bx{\b{x}}
\newcommand\bI{\b{I}}
\newcommand\N{{\cal N}}
\newcommand\D{{\cal D}}
\newcommand\T{\mathrm T}
\]
\[
\begin{align}
p(\D\mid\alpha,\beta) = \int p(\D\mid\bw,\beta)p(\bw\mid\alpha)\,d\bw \tag{5.174}
\end{align}
\]
において、\(f(\bw)=p(\D\mid\bw,\beta)p(\bw\mid\alpha),\ Z=p(\D\mid\alpha,\beta) \) として、(4.135)を適用すると
\[
Z\simeq f(\bw_{MAP})\frac{(2\pi)^{W\over 2}}{|\bA|^{1 \over 2}},\ \ \ (W\ は\ \bw\ の次元)
\]
となる。ここで
\[
\begin{align}
f(\bw_{MAP}) &= p(\D\mid\bw_{MAP},\beta)p(\bw_{MAP}\mid\alpha) \\
&= \prod_{n=1}^N \underset{(\because\ (5.163))}{\N(t_n\mid y(\bx_n,\bw_{MAP}),\beta^{-1})}
\underset{(\because\ (5.162))}{\N(\bw_{MAP}\mid\b{0},\alpha^{-1}\bI)} \\
&= \prod_{n=1}^N {1\over (2\pi)^{1/2}}{1\over (\beta^{-1})^{1/2}}\exp\l[-{1\over 2\beta^{-1}}\{t_n-y(\bx_n,\bw_{MAP})\}^2\r] \\
&~~~~~{1\over (2\pi)^{W/2}}{1\over |\alpha^{-1}\bI|^{1/2}}\exp\l\{-{1\over 2}\bw_{MAP}^\T(\alpha^{-1}\bI)^{-1}\bw_{MAP}\r\} \\
&= \prod_{n=1}^N \l({\beta\over 2\pi}\r)^{1/2}\exp\l[-{\beta\over 2}\{t_n-y(\bx_n,\bw_{MAP})\}^2\r] \\
&~~~~~\underset{\cmt{※1}}{\l({\alpha\over 2\pi}\r)^{W\over 2}}
\underset{\cmt{※2}}{\exp\l(-{\alpha\over 2}\bw_{MAP}^\T\bw_{MAP}\r)}
\end{align}
\]
なので
\[
\begin{align}
\ln p(\D\mid\alpha,\beta)
&\simeq \ln f(\bw_{MAP})+{W\over 2}\ln(2\pi)-{1\over 2}\ln|\bA| \\
&= \sum_{n=1}^N\l[{1\over2}\{\ln\beta-\ln(2\pi)\}-{\beta\over2}\{t_n-y(\bx_n,\bw_{MAP})\}^2\r] \\
&~~~~~ + {W\over2}\{\ln\alpha-\ln(2\pi)\}-{\alpha\over2}\bw_{MAP}^\T\bw_{MAP}+{W\over2}\ln(2\pi)-{1\over2}\ln|\bA| \\
&=-\l[{\beta\over2}\sum_{n=1}^N\{t_n-y(\bx_n,\bw_{MAP})\}^2+{\alpha\over2}\bw_{MAP}^\T\bw_{MAP}\r] \\
&~~~~~ -{1\over2}\ln|\bA|+{N\over2}\ln\beta-{N\over2}\ln(2\pi)+{W\over2}\ln\alpha \\
&=-E(\bw_{MAP})-{1\over2}\ln|\bA|+{N\over2}\ln\beta-{N\over2}\ln(2\pi)+{W\over2}\ln\alpha \tag{5.175}
\end{align}
\]
を得る。ただし
\[
\begin{align}
E(\bw_{MAP})={\beta\over2}\sum_{n=1}^N\{t_n-y(\bx_n,\bw_{MAP})\}^2+{\alpha\over2}\bw_{MAP}^\T\bw_{MAP} \tag{5.176}
\end{align}
\]
とする。