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 .
Theorems[edit]
generalization of chi square-Dirichlet result[edit]
Suppose are independently distributed Wishart positive definite matrices. Then, defining (where is the sum of the matrices and is any reasonable factorization of ), we have
Marginal distribution[edit]
If , and if , then:
Conditional distribution[edit]
Also, with the same notation as above, the density of is given by
where we write .
partitioned distribution[edit]
Suppose and suppose that is a partition of (that is, and if ). Then, writing and (with ), we have:
partitions[edit]
Suppose . Define
where is and is . Writing the Schur complement we have
and
See also[edit]
References[edit]
A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.