In statistics, the inverse Dirichlet distribution is a derivation of the matrix variate Dirichlet distribution. It is related to the inverse Wishart distribution.
Suppose
are
positive definite matrices with a matrix variate Dirichlet distribution,
. Then
have an inverse Dirichlet distribution, written
. 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af957d93d8a5a93a5752667c64c1947b0d5f2d37)
A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.