Inverse Dirichlet distribution explained

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 x 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} ^{-1} ^{i}} ,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 ^{- a _{i-(p+1)/2}} _{i\right)} \det\left(I _{p-\sum} ^{-1} ^{i}} \right) ^{a _{r+1} - (p +1 )/ 2}

References

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