Matrix variate Dirichlet distribution

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

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generalization of chi square-Dirichlet result

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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

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If  , and if  , then:

 

Conditional distribution

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Also, with the same notation as above, the density of   is given by

 

where we write  .

partitioned distribution

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Suppose   and suppose that   is a partition of   (that is,   and   if  ). Then, writing   and   (with  ), we have:

 

partitions

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Suppose  . Define

 

where   is   and   is  . Writing the Schur complement   we have

 

and

 

See also

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References

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