# 逆威沙特分佈

參數 $m > p-1\!$ 自由度 (實數) $\mathbf{\Psi} > 0\,$ 尺度矩陣 (正定) $\mathbf{W}\!$是正定的 $\frac{\left|{\mathbf\Psi}\right|^{m/2}\left|B\right|^{-(m+p+1)/2}e^{-\mathrm{trace}({\mathbf\Psi}{\mathbf B}^{-1})/2} }{2^{mp/2}\Gamma_p(m/2)}$ $\frac{\mathbf{\Psi}}{m - p - 1}$ $\frac{\mathbf{\Psi}}{m + p + 1}$[1]:406

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

## 概率密度函数

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

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

## 相关定理

### 威沙特分布矩阵之逆的概率分布

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

### 威沙特分布矩阵之逆的边际与条件分布

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

### 矩相关特性

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

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

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

## 相关分布

$p(x|\alpha, \beta) = \frac{\beta^\alpha\, x^{-\alpha-1} \exp(-\beta/x)}{\Gamma_1(\alpha)}.$

## 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. ^ 2.0 2.1 Kanti V. Mardia, J. T. Kent and J. M. Bibby. Multivariate Analysis. Academic Press. 1979. ISBN 0-12-471250-9.