多元高斯分布：
$$
x \sim N(\mu, \Sigma) = \frac{1}{ { (2 \pi)^{\frac{p}{2}} | {\Sigma}|^{\frac{1}{2}}}} \exp \left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^{\mathrm{T}} \mathbf{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right)
$$
$x$为随机变量，$x \in \mathbb{R}^p$
$$
x=\left(\begin{array}{c}
{x_{1}} \\
{x_{2}} \\
{\vdots} \\
{x_{p}}
\end{array}\right) \quad \mu=\left(\begin{array}{c}
{\mu_{1}} \\
{\mu_{2}} \\
{\vdots} \\
{\mu_{p}}
\end{array}\right) \quad \Sigma=\left(\begin{array}{ccc}
{\sigma_{11}} & {\sigma_{12}} & {\cdots \sigma_{1p}} \\
{\sigma_{21}} & {\sigma_{22}} & {\cdots \sigma_{2 p}} \\
{\vdots} & {\vdots} & {\vdots} \\
{\sigma_{p1}} & {\sigma_{p 2} \cdots} & {\sigma_{p p}}
\end{array}\right)_{p\times p}
$$
已知：
$$
x = \left(\begin{array}{l}
{x_{a}} \\
{x_{b}}
\end{array}\right)_{\rightarrow\ n}^{\rightarrow\ m}
\quad
m + n = p
$$
$$
\mu=\left(\begin{array}{l}
{\mu_{a}} \\
{\mu_{b}}
\end{array}\right)
\quad
\Sigma=\left(\begin{array}{l}
{\Sigma_{a a}\ \Sigma_{a b}} \\
{\Sigma_{b a}\ \Sigma_{b b}}
\end{array}\right)
$$
求：$p\left(x_{a}\right), p\left(x_{b} | x_{a}\right)$ 或 $p\left(x_{b}\right), p\left(x_{a} | x_{b}\right)$
$$
\begin{aligned}
&x_{a}= \left(\begin{array}{ll}
{I_{m}} & {0_{n}}
\end{array}\right)\left(\begin{array}{l}
{x_{a}} \\
{x_{b}}
\end{array}\right)
\end{aligned}
$$
$$
E\left[x_{a}\right]=\left(\begin{array}{ll}
{I_{m}} & {0_{n}}
\end{array}\right)\left(\begin{array}{l}
{\mu_{a}} \\
{\mu_{b}}
\end{array}\right)=\mu_{a}
$$
\begin{aligned}
Var\left[x_{a}\right] &=\left(\begin{array}{ll}
{I_m } & {0}
\end{array}\right)\left(\begin{array}{l}
{\Sigma_{a a}\ \Sigma_{a b}} \\
{\Sigma_{b a}\ \Sigma_{b b}}
\end{array}\right)\left(\begin{array}{l}
{I_m } \\
{0}
\end{array}\right) \\
&=\Sigma_{a a}
\end{aligned}
$$
x_{a} \sim N\left(\mu_{a}, \Sigma_{a a} \right)
$$
下面求$x_b|x_a$
令\begin{cases} x_{b \cdot a}=x_{b}-\Sigma_{b a} \Sigma_{a a}^{-1} x_{a} \\ \mu_{b \cdot a}=\mu_{b}-\Sigma_{b a} \Sigma_{a a}^{-1} \mu_{a}\\ \Sigma_{b b \cdot a}=\Sigma_{b b}-\Sigma_{b a} \Sigma_{a a}^{-1} \Sigma_{a b} \end{cases}
$$
x_{b \cdot a}=\left(\begin{array}{ll}
{-\Sigma_{b a} \Sigma_{a a}^{-1}} & {I_{n}}\end{array}\right)\left(\begin{array}{l}
{x_{a}} \\
{x_{b}}
\end{array}\right)
$$
$$
E\left[x_{b \cdot a}\right]=\left(\begin{array}{ll}
{-\Sigma_{b a} \Sigma_{a a}^{-1}} & {I_{n}}\end{array}\right) \cdot\left(\begin{array}{l}
{\mu_{a}} \\
{\mu_{b}}
\end{array}\right)=\mu_{b}-\Sigma_{b a} \Sigma_{a a}^{-1} \mu_{a}=\mu_{b\cdot a}
$$
$$
\begin{aligned}
\operatorname{Var}\left[x_{b\cdot a}\right]=\left(\begin{array}{ll}
{-\Sigma_{b a} \Sigma_{a a}^{-1}} & {I_{n}}\end{array}\right)\left(\begin{array}{c}
{\Sigma_{a a}\ \Sigma_{a b}} \\
{\Sigma_{b a}\ \Sigma_{b b}}
\end{array}\right)\left(\begin{array}{cc}
{-\Sigma_{a a}^{-1} \Sigma_{b a}^T } \\ {I_{n}} \end{array}\right)=\Sigma_{b b}-\Sigma_{b a} \Sigma_{a a}^{-1} \Sigma_{a b}=\Sigma_{b b \cdot a}
\end{aligned}
$$
$$x_{b\cdot a} \sim N\left(\mu_{b \cdot a}, \Sigma_{bb\cdot a} \right)\\$$
$$x_{b}=x_{b \cdot a} + \Sigma_{b a} \Sigma_{a a}^{-1} x_{a}$$
$$
\begin{aligned}
&E\left[x_{b} | x_{a}\right]=\mu_{b \cdot a} + \Sigma_{b a} \Sigma_{a a}^{-1} x_{a}\\
&\operatorname{Var}\left[x_{b} | x_{a}\right]=\operatorname{Var}\left[x_{b\cdot a} \right]=\Sigma_{b b \cdot a}
\end{aligned}
$$
$$ x_b | x_a \sim N \left(\mu_{b \cdot a} + \Sigma_{b a} \Sigma_{a a}^{-1} x_{a}, \Sigma_{b b \cdot a} \right) $$
补充：$x_{b\cdot a}$与$x_a$独立：
