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 $$I_p-\sum_{i=1}^rU_i$$ 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, $$\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_r;a_{r+1}\right)$$, if their joint probability density function is



\left\{\beta_p\left(a_1,\ldots,a_r,a_{r+1}\right)\right\}^{-1}\prod_{i=1}^{r}\det\left(U_i\right)^{a_i-(p+1)/2}\det\left(I_p-\sum_{i=1}^rU_i\right)^{a_{r+1}-(p+1)/2} $$

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 $$U_{r+1}=I_p-\sum_{i=1}^r U_i$$ then the PDF takes the simpler form



\left\{\beta_p\left(a_1,\ldots,a_{r+1}\right)\right\}^{-1}\prod_{i=1}^{r+1}\det\left(U_i\right)^{a_i-(p+1)/2}, $$

on the understanding that $$\sum_{i=1}^{r+1}U_i=I_p$$.

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 $$U_i=S^{-1/2}S_i\left(S^{-1/2}\right)^T$$ (where $$S=\sum_{i=1}^{r+1}S_i$$ is the sum of the matrices and $$S^{1/2}\left(S^{-1/2}\right)^T$$ is any reasonable factorization of $$S$$), we have



\left(U_1,\ldots,U_r\right)\sim D_p\left(n_1/2,...,n_{r+1}/2\right). $$

Marginal distribution
If $$\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_{r+1}\right)$$, and if $$s\leq r$$, then:



\left(U_1,\ldots,U_s\right)\sim D_p\left(a_1,\ldots,a_s,\sum_{i=s+1}^{r+1}a_i\right) $$

Conditional distribution
Also, with the same notation as above, the density of $$\left(U_{s+1},\ldots,U_r\right)\left|\left(U_1,\ldots,U_s\right)\right.$$ is given by



\frac{ \prod_{i=s+1}^{r+1}\det\left(U_i\right)^{a_i-(p+1)/2} }{ \beta_p\left(a_{s+1},\ldots,a_{r+1}\right)\det\left(I_p-\sum_{i=1}^{s}U_i\right)^{\sum_{i=s+1}^{r+1}a_i-(p+1)/2} } $$ where we write $$U_{r+1} = I_p-\sum_{i=1}^rU_i$$.

partitioned distribution
Suppose $$\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_{r+1}\right)$$ and suppose that $$S_1,\ldots,S_t$$ is a partition of $$\left[r+1\right]=\left\{1,\ldots r+1\right\}$$ (that is, $$\cup_{i=1}^tS_i=\left[r+1\right]$$ and $$S_i\cap S_j=\emptyset$$ if $$i\neq j$$). Then, writing $$U_{(j)}=\sum_{i\in S_j}U_i$$ and $$a_{(j)}=\sum_{i\in S_j}a_i$$ (with $$U_{r+1}=I_p-\sum_{i=1}^r U_r$$), we have:



\left(U_{(1)},\ldots U_{(t)}\right)\sim D_p\left(a_{(1)},\ldots,a_{(t)}\right).$$

partitions
Suppose $$\left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_{r+1}\right)$$. Define
 * $$U_i=

\left( \begin{array}{rr} U_{11(i)} & U_{12(i)} \\ U_{21(i)} & U_{22(i)} \end{array} \right) \qquad i=1,\ldots,r $$

where $$U_{11(i)}$$ is $$p_1\times p_1$$ and $$U_{22(i)}$$ is $$p_2\times p_2$$. Writing the Schur complement $$U_{22\cdot 1(i)} = U_{21(i)} U_{11(i)}^{-1}U_{12(i)}$$ we have



\left(U_{11(1)},\ldots,U_{11(r)}\right)\sim D_{p_1}\left(a_1,\ldots,a_{r+1}\right)$$ and



\left(U_{22.1(1)},\ldots,U_{22.1(r)}\right)\sim D_{p_2}\left(a_1-p_1/2,\ldots,a_r-p_1/2,a_{r+1}-p_1/2+p_1r/2\right). $$