PRML 1章 メモ

Math

(1.45)

\int p(D|{\mathrm w})p({\mathrm w})d{\mathrm w} = \int p(D,{\mathrm w})d{\mathrm w}

加法定理より

= p(D)

(1.54)

\displaystyle \ln{p({\sf x}|\mu,\sigma)} = \ln{\prod_{n=1}^{N}\frac{1}{(2\pi\sigma^{2})^{\frac{1}{2}}}\exp{\{-\frac{1}{2\sigma^{2}}(x_n-\mu)^{2}}}\}

\displaystyle =\sum_{n=1}^{N}\ln{\frac{1}{(2\pi\sigma^{2})^\frac{1}{2}}\exp{\{-\frac{1}{2\sigma^{2}}(x_n-\mu)^{2}}}\}

\displaystyle =-\frac{1}{2\sigma^{2}}\sum_{n=1}^{N}(x_n-\mu)^{2} - \sum_{n=1}^{N}\ln{(2\pi\sigma^{2})^{\frac{1}{2}}}

\displaystyle =-\frac{1}{2\sigma^{2}}\sum_{n=1}^{N}(x_n-\mu)^{2} - \frac{N}{2}\ln{2\pi\sigma^{2}}

\displaystyle =-\frac{1}{2\sigma^{2}}\sum_{n=1}^{N}(x_n-\mu)^{2} - \frac{N}{2}\ln{\sigma^{2}} - \frac{N}{2}\ln{(2\pi)}