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

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

References

A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.