In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution.
Suppose are positive definite matrices with also positive-definite, where is the identity matrix. Then we say that the have a matrix variate Dirichlet distribution, , if their joint probability density function is
where and is the multivariate beta function.
If we write then the PDF takes the simpler form
on the understanding that .
Suppose are independently distributed Wishart positive definite matrices. Then, defining (where <Math>S=\sum_{i=1}^{r+1}S_i</math> is the sum of the matrices and is any reasonable factorization of ), we have
If , and if , then:
Also, with the same notation as above, the density of is given by
where we write .
Suppose and suppose that is a partition of (that is, and if ). Then, writing and (with ), we have:
Suppose . Define
where is and is . Writing the Schur complement we have
and
A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.