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

[math]\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} }[/math]

References

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