Jump to content

Matrix variate Dirichlet distribution

From Wikipedia, the free encyclopedia

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.