后面推导中反复用到Woodbury恒等式 \[ (A+UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1} \]

1. 边缘分布

假设 \(\boldsymbol{x}\sim {\mathcal N}(\boldsymbol{\mu}, \boldsymbol\Sigma_1), \boldsymbol{y}|\boldsymbol{x}\sim\mathcal{N}(\boldsymbol{A}\boldsymbol{x}, \boldsymbol\Sigma_2)\)，那么 \(\boldsymbol{y}\) 的边缘分布也是高斯分布 \[ \boldsymbol{y} \sim \mathcal{N}(\boldsymbol{\mu}_3, \boldsymbol\Sigma_3) \\ \boldsymbol\Sigma_3 = (\boldsymbol\Sigma_2^{-1} - \boldsymbol\Sigma_2^{-1}\boldsymbol{A} (\boldsymbol\Sigma_1^{-1}+\boldsymbol{A}^{\rm T} \boldsymbol\Sigma_2^{-1} \boldsymbol{A})^{-1} \boldsymbol{A}^{\rm T} \boldsymbol\Sigma_2^{-1})^{-1} = \boldsymbol\Sigma_2 + \boldsymbol{A}\boldsymbol\Sigma_1 \boldsymbol{A}^{\rm T} \\ \boldsymbol{\mu}_3 = \boldsymbol\Sigma_3 \boldsymbol\Sigma_2^{-\rm T} \boldsymbol{A} (\boldsymbol\Sigma_1^{-1}+\boldsymbol{A}^{\rm T} \boldsymbol\Sigma_2^{-1} \boldsymbol{A})^{-1} \boldsymbol\Sigma_1^{-1} \boldsymbol{\mu} = \boldsymbol{A\mu}\\ \]

2. 后验分布

假设有先验 \(\boldsymbol{x}\sim {\mathcal N}(\boldsymbol{\mu}, \boldsymbol\Sigma_1)\)，似然 \(\boldsymbol{y}|\boldsymbol{x}\sim\mathcal{N}(\boldsymbol{A}\boldsymbol{x}, \boldsymbol\Sigma_2)\)，那么关于 \(\boldsymbol{x}\) 的后验分布也是高斯分布 \[ p(\boldsymbol{x}| \hat{\boldsymbol{y}}) \propto p(\boldsymbol{x},\hat{\boldsymbol{y}}) = p(\boldsymbol{x}) p(\hat{\boldsymbol{y}} | \boldsymbol{x}) \sim {\mathcal N}(\boldsymbol{\mu}_4, \boldsymbol\Sigma_4) \\ \boldsymbol{\Sigma}_4 = (\boldsymbol\Sigma_1^{-1} + \boldsymbol{A}^{\rm T}\boldsymbol{\Sigma}_2^{-1}\boldsymbol{A})^{-1} \\ \boldsymbol{\mu}_4 = \boldsymbol{\Sigma}_4 (\boldsymbol{\Sigma}_1^{-1}\boldsymbol{\mu} + \boldsymbol{A}^{\rm T}\boldsymbol{\Sigma}_2^{-1}\hat{\boldsymbol{y}}) \] 特殊情况下，假如 \(\boldsymbol{A}=\boldsymbol{I}, \boldsymbol{\Sigma}_2=\sigma^2\boldsymbol{I},\boldsymbol{\mu}=\boldsymbol{0}\)，此时将有 \[ \boldsymbol{\Sigma}_4 = \sigma^2 \boldsymbol{\Sigma}_1 (\boldsymbol{\Sigma}_1 + \sigma^2\boldsymbol{I})^{-1} \\ \boldsymbol\mu_4 = \boldsymbol{\Sigma}_1(\boldsymbol{\Sigma}_1+\sigma^2\boldsymbol{I})^{-1}\hat{\boldsymbol{y}} \]

3. 条件分布

假设 \(\boldsymbol{x}_1,\boldsymbol{x}_2\) 服从联合高斯分布 \[ \begin{bmatrix} \boldsymbol{x}_1 \\ \boldsymbol{x}_2 \end{bmatrix} \sim {\mathcal N} \left( \begin{bmatrix} \boldsymbol{\mu}_1 \\ \boldsymbol{\mu}_2 \end{bmatrix}, \begin{bmatrix} \boldsymbol{\Sigma}_{11} & \boldsymbol{\Sigma}_{12} \\ \boldsymbol{\Sigma}_{21} & \boldsymbol{\Sigma}_{22} \end{bmatrix} \right) \] 那么 \(\boldsymbol{x}_2 | \boldsymbol{x}_1\) 也服从高斯分布 \[ \boldsymbol{x}_2 | \boldsymbol{x}_1 \sim {\mathcal N}(\boldsymbol{\mu}_5, \boldsymbol{\Sigma}_5) \\ \boldsymbol{\Sigma}_5 = \boldsymbol{\Sigma}_{22} - \boldsymbol{\Sigma}_{21} \boldsymbol{\Sigma}_{11}^{-1} \boldsymbol{\Sigma}_{12} \\ \boldsymbol{\mu}_5 = \boldsymbol{\mu}_2 + \boldsymbol{\Sigma}_{21} \boldsymbol{\Sigma}_{11}^{-1} (\hat{\boldsymbol{x}}_1 - \boldsymbol{\mu}_1) \]