ガウス分布関数

一次元

(1)\[\mathcal{N}(x|\mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp{\left( -\frac{(x-\mu)^2}{2\sigma^2} \right)}\]

N次元

(2)\[\mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}, \boldsymbol{\Sigma}) = \frac{1}{\sqrt{(2\pi)^N\left|\boldsymbol{\Sigma}\right|}} \exp{\left( -\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\boldsymbol{x}-\boldsymbol{\mu}) \right)}\]