Inverse Dirichlet distribution

In statistics, the inverse Dirichlet distribution is a derivation of the matrix variate Dirichlet distribution. It is related to the inverse Wishart distribution.

Suppose ${\displaystyle U_{1},\ldots ,U_{r}}$ are ${\displaystyle p\times p}$ positive definite matrices with a matrix variate Dirichlet distribution, ${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)}$. Then ${\displaystyle X_{i}={U_{i}}^{-1},i=1,\ldots ,r}$ have an inverse Dirichlet distribution, written ${\displaystyle \left(X_{1},\ldots ,X_{r}\right)\sim \operatorname {ID} \left(a_{1},\ldots ,a_{r};a_{r+1}\right)}$. Their joint probability density function is given by

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