Matrix variate Dirichlet distribution

In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution.

Suppose $U_{1},\ldots ,U_{r}$ are $p\times p$ positive definite matrices with $\sum _{i=1}^{r}U_{i}<I_{p}$ , where $I_{p}$ is the $p\times p$ identity matrix. Then we say that the $U_{i}$ have a matrix variate Dirichlet distribution, $\left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)$ , if their joint probability density function is

\left\{\beta _{p}\left(a_{1},\ldots ,a_{r},a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r}\det \left(U_{i}\right)^{a_{i}-(p+1)/2}\det \left(I_{p}-\sum _{i=1}^{r}U_{i}\right)^{a_{r+1}-(p+1)/2}

where $a_{i}>(p-1)/2,i=1,\ldots ,r+1$ and $\beta _{p}\left(\cdots \right)$ is the multivariate beta function.

If we write $U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{i}$ then the PDF takes the simpler form

\left\{\beta _{p}\left(a_{1},\ldots ,a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r+1}\det \left(U_{i}\right)^{a_{i}-(p+1)/2},

on the understanding that $\sum _{i=1}^{r}U_{i}=I_{p}$ .

Theorem

Suppose $S_{i}\sim W_{p}\left(n_{i},\Sigma \right),i=1,\ldots ,r+1$ are independently distributed Wishart $p\times p$ positive definite matrices. Then, defining $U_{i}=S^{-1/2}S_{i}\left(S^{-1/2}\right)^{T}$ (where $S=\sum _{i=1}^{r+1}S_{i}$ is the sum of the matrices and $S^{1/2}\left(S^{-1/2}\right)^{T}$ is any reasonable factorization of $S$ ), we have