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 $U_1,\dots ,U_r$ are $p\times p$ positive definite matrices with a matrix variate Dirichlet distribution, $(U_1,\dots ,U_r)\sim D_p(a_1,\dots ,a_r;a_{r+1})$. Then $X_i={U_i}^{-1},i=1,\dots ,r$ have an inverse Dirichlet distribution, written $(X_1,\dots ,X_r)\sim \mathrm{ID}(a_1,\dots ,a_r;a_{r+1})$. Their joint probability density function is given by

${\left\{{\beta }_p(a_1,\dots ,a_r;a_{r+1})\right\}}^{-1}\prod _{i=1}^r\det {\left(X_i\right)}^{-a_i-(p+1)/2}\det {(I_p-\sum _{i=1}^r{X_i}^{-1})}^{a_{r+1}-(p+1)/2}$

References

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