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 $U\_1,\backslash ldots,U\_r$ are $p\backslash times\; p$ positive definite matrices with a matrix variate Dirichlet distribution, $\backslash left(U\_1,\backslash ldots,U\_r\backslash right)\backslash sim\; D\_p\backslash left(a\_1,\backslash ldots,a\_r;a\_\{r+1\}\backslash right)$. Then $X\_i=\{U\_i\}^\{-1\},i=1,\backslash ldots,r$ have an inverse Dirichlet distribution, written $\backslash left(X\_1,\backslash ldots,X\_r\backslash right)\backslash sim\; \backslash operatorname\{ID\}\backslash left(a\_1,\backslash ldots,a\_r;a\_\{r+1\}\backslash right)$. 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}