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 $U_1,\ldots,U_r$ are $p\times p$ positive definite matrices with $I_p-\sum_^rU_i$ also positive-definite, where $I_p$ is the $p\times p$ identity matrix. Then we say that the $U_i$ have a matrix variate Dirichlet distribution, $\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_r;a_\right)$, if their joint probability density function is

:$\left\^\prod_^\det\left(U_i\right)^\det\left(I_p-\sum_^rU_i\right)^$

where $a_i>(p-1)/2,i=1,\ldots,r+1$ and $\beta_p\left(\cdots\right)$ is the multivariate beta function.

If we write $U_=I_p-\sum_^r U_i$ then the PDF takes the simpler form

:$\left\^\prod_^\det\left(U_i\right)^,$

on the understanding that $\sum_^U_i=I_p$.

Theorems

generalization of chi square-Dirichlet result

Suppose $S_i\sim W_p\left(n_i,\Sigma\right),i=1,\ldots,r+1$ are independently distributed Wishart $p\times p$ positive definite matrices. Then, defining $U_i=S^S_i\left(S^\right)^T$ (where $S=\sum_^S_i$ is the sum of the matrices and $S^\left(S^\right)^T$ is any reasonable factorization of $S$), we have

:$\left(U_1,\ldots,U_r\right)\sim D_p\left(n_1/2,...,n_/2\right).$

Marginal distribution

If $\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_\right)$, and if $s\leq r$, then:

:$\left(U_1,\ldots,U_s\right)\sim D_p\left(a_1,\ldots,a_s,\sum_^a_i\right)$

Conditional distribution

Also, with the same notation as above, the density of $\left(U_,\ldots,U_r\right)\left, \left(U_1,\ldots,U_s\right)\right.$ is given by

:$\frac$
where we write $U_ = I_p-\sum_^rU_i$.

partitioned distribution

Suppose $\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_\right)$ and suppose that $S_1,\ldots,S_t$ is a partition of \left +1\right\left\ (that is, \cup_^tS_i=\left +1\right/math> and $S_i\cap S_j=\emptyset$ if $i\neq j$). Then, writing $U_=\sum_U_i$ and $a_=\sum_a_i$ (with $U_=I_p-\sum_^r U_r$), we have:

:$\left(U_,\ldots U_\right)\sim D_p\left(a_,\ldots,a_\right).$

partitions

Suppose $\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_\right)$. Define
:$U_i= \left( \begin U_ & U_ \\ U_ & U_ \end \right) \qquad i=1,\ldots,r$

where $U_$ is $p_1\times p_1$ and $U_$ is $p_2\times p_2$. Writing the Schur complement $U_ = U_ U_^U_$ we have

:$\left(U_,\ldots,U_\right)\sim D_\left(a_1,\ldots,a_\right)$
and

:$\left(U_,\ldots,U_\right)\sim D_\left(a_1-p_1/2,\ldots,a_r-p_1/2,a_-p_1/2+p_1r/2\right).$

See also

* Inverse Dirichlet distribution

References

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

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Probability distributions