多変量正規分布

互いに独立ではない１組の計測値の確率モデル。

2次元

\begin{aligned} f (x_{1}, x_{2}; μ_{1}, μ_{2}, σ_{1}^{2}, σ_{2}^{2}, ρ_{12}) \\ = & \frac{1}{2 π \sqrt{σ_{1}^{2} σ_{2}^{2} - ρ_{12}^{2} σ_{1}^{2} σ_{2}^{2}}} \exp {- \frac{({(x_{1} - μ_{1})}^{2} σ_{2}^{2} + {(x_{2} - μ_{2})}^{2} σ_{1}^{2} - 2 (x_{1} - μ_{1}) (x_{2} - μ_{2}) ρ_{12} σ_{1} σ_{2})}{2 (σ_{1}^{2} σ_{2}^{2} - ρ_{12}^{2} σ_{1}^{2} σ_{2}^{2})}} \\ = & \frac{1}{2 π σ_{1} σ_{2} \sqrt{1 - ρ_{12}^{2}}} \exp {- \frac{({(x_{1} - μ_{1})}^{2} σ_{2}^{2} + {(x_{2} - μ_{2})}^{2} σ_{1}^{2} - 2 (x_{1} - μ_{1}) (x_{2} - μ_{2}) ρ_{12} σ_{1} σ_{2})}{2 σ_{1}^{2} σ_{2}^{2} (1 - ρ_{12}^{2})}} \end{aligned}

\boldsymbol μ = (\begin{matrix} μ_{1} \\ μ_{2} \\ ⋮ \\ μ_{p} \end{matrix})

Σ = (\begin{array}{cccc} σ_{1}^{2} & ρ_{12} σ_{1} σ_{2} & \dots & ρ_{1 p} σ_{1} σ_{p} \\ ρ_{21} σ_{2} σ_{1} & σ_{2}^{2} & \dots & ρ_{2 p} σ_{2} σ_{p} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ρ_{p 1} σ_{p} σ_{1} & ρ_{p 2} σ_{p} σ_{2} & \dots & σ_{p}^{2} \end{array})

\boldsymbol X = (\begin{matrix} X_{1} \\ X_{2} \\ ⋮ \\ X_{p} \end{matrix}) \sim N (\boldsymbol μ, Σ)

f (\boldsymbol x; \boldsymbol μ, Σ) = \frac{1}{\sqrt{| 2 π Σ |}} \exp {- \frac{1}{2} {(\boldsymbol x - \boldsymbol μ)}^{⊤} Σ (\boldsymbol x - \boldsymbol μ)}