PRML-公式推导 - 1.67公式详解

$ p(w|X, T, \alpha, \beta) \propto p(T|X, w, \beta)p(w|\alpha)$

\(取ln\)
\(\ln p(T|X, w, \beta) + \ln p(w|\alpha)\)

回顾
$ \ln p(T|X, w, \beta) = -\frac{\beta}{2}\sum\limits_{n=1}^{N}{y(x_n, w) - t_n}^2 + \frac{N}{2}\ln{\beta} - \frac{N}{2}\ln{(2\pi)} \tag{1.62} $

\(p(\boldsymbol{w} | \alpha)=\mathcal{N}\left(\boldsymbol{w} | \mathbf{0}, \alpha^{-1} \boldsymbol{I}\right)=\left(\frac{\alpha}{2 \pi}\right)^{\frac{M+1}{2}} \exp \left\{-\frac{\alpha}{2} \boldsymbol{w}^{T} \boldsymbol{w}\right\} \tag{1.65}\)

\(则\ln p(T|X, w, \beta) + \ln p(w|\alpha)\)
\(=-\frac{\beta}{2}\sum\limits_{n=1}^{N}\{y(x_n, w) - t_n\}^2 + \frac{N}{2}\ln{\beta} - \frac{N}{2}\ln{(2\pi)}-\frac{\alpha}{2}{w}^{T}{w}+\frac{M+1}{2} \ln \frac{\alpha}{2\pi}\)
\(其中只有-\frac{\beta}{2}\sum\limits_{n=1}^{N}\{y(x_n, w) - t_n\}^2-\frac{\alpha}{2}{w}^{T}{w} 与w有关\)
\(所以原式=-\frac{\beta}{2}\sum\limits_{n=1}^{N}\{y(x_n, w) - t_n\}^2-\frac{\alpha}{2}{w}^{T}{w} +C\)