# 逆威沙特分布

参数 ${\displaystyle m>p-1\!}$ 自由度 (实数)${\displaystyle \mathbf {\Psi } >0\,}$ 尺度矩阵 (正定) ${\displaystyle \mathbf {W} \!}$是正定的 ${\displaystyle {\frac {\left|{\mathbf {\Psi } }\right|^{m/2}\left|B\right|^{-(m+p+1)/2}e^{-\mathrm {trace} ({\mathbf {\Psi } }{\mathbf {B} }^{-1})/2}}{2^{mp/2}\Gamma _{p}(m/2)}}}$ ${\displaystyle {\frac {\mathbf {\Psi } }{m-p-1}}}$ ${\displaystyle {\frac {\mathbf {\Psi } }{m+p+1}}}$:406

${\displaystyle \mathbf {B} \sim W^{-1}({\mathbf {\Psi } },m)}$

## 概率密度函数

${\displaystyle {\frac {\left|{\mathbf {\Psi } }\right|^{m/2}\left|\mathbf {B} \right|^{-(m+p+1)/2}e^{-\mathrm {trace} ({\mathbf {\Psi } }{\mathbf {B} }^{-1})/2}}{2^{mp/2}\Gamma _{p}(m/2)}},}$

${\displaystyle \mathrm {trace} \;:\quad \mathbf {M} \quad \rightarrow \quad \mathrm {trace} (\mathbf {M} )}$

## 相关定理

### 威沙特分布矩阵之逆的概率分布

${\displaystyle p(\mathbf {B} |\mathbf {\Psi } ,m)={\frac {\left|{\mathbf {\Psi } }\right|^{m/2}\left|\mathbf {B} \right|^{-(m+p+1)/2}\exp \left({-\mathrm {tr} ({\mathbf {\Psi } }{\mathbf {B} }^{-1})/2}\right)}{2^{mp/2}\Gamma _{p}(m/2)}}}$

### 威沙特分布矩阵之逆的边际与条件分布

${\displaystyle {\mathbf {A} }={\begin{bmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}\\\mathbf {A} _{21}&\mathbf {A} _{22}\end{bmatrix}},\;{\mathbf {\Psi } }={\begin{bmatrix}\mathbf {\Psi } _{11}&\mathbf {\Psi } _{12}\\\mathbf {\Psi } _{21}&\mathbf {\Psi } _{22}\end{bmatrix}}}$

### 矩相关特性

${\displaystyle E(\mathbf {B} )={\frac {\mathbf {\Psi } }{m-p-1}}.}$

${\displaystyle {\mbox{var}}(b_{ij})={\frac {(m-p+1)\psi _{ij}^{2}+(m-p-1)\psi _{ii}\psi _{jj}}{(m-p)(m-p-1)^{2}(m-p-3)}}}$

${\displaystyle {\mbox{var}}(b_{ii})={\frac {2\psi _{ii}^{2}}{(m-p-1)^{2}(m-p-3)}}.}$

## 相关分布

${\displaystyle p(x|\alpha ,\beta )={\frac {\beta ^{\alpha }\,x^{-\alpha -1}\exp(-\beta /x)}{\Gamma _{1}(\alpha )}}.}$

## 参考来源

1. ^ A. O'Hagan, and J. J. Forster. Kendall's Advanced Theory of Statistics: Bayesian Inference 2B 2. Arnold. 2004. ISBN 0-340-80752-0.
2. Kanti V. Mardia, J. T. Kent and J. M. Bibby. Multivariate Analysis. Academic Press. 1979. ISBN 0-12-471250-9.