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 $U_1,\dots ,U_r$ are $p\times p$ positive definite matrices with $I_p-\sum _{i=1}^r{U}_i$ 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, $(U_1,\dots ,U_r)\sim D_p(a_1,\dots ,a_r;a_{r+1})$, if their joint probability density function is

${\left\{{\beta }_p(a_1,\dots ,a_r,a_{r+1})\right\}}^{-1}\prod _{i=1}^r\det {\left(U_i\right)}^{a_i-(p+1)/2}\det {(I_p-\sum _{i=1}^r{U}_i)}^{a_{r+1}-(p+1)/2}$

where $a_i>(p-1)/2,i=1,\dots ,r+1$ and ${\beta }_p(\cdots )$ 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

${\left\{{\beta }_p(a_1,\dots ,a_{r+1})\right\}}^{-1}\prod _{i=1}^{r+1}\det {\left(U_i\right)}^{a_i-(p+1)/2},$

on the understanding that $\sum _{i=1}^{r+1}{U}_i=I_p$.

Theorems

generalization of chi square-Dirichlet result

Suppose $S_i\sim W_p(n_i,\Sigma ),i=1,\dots ,r+1$ 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

$(U_1,\dots ,U_r)\sim D_p(n_1/2,\mathrm{...},n_{r+1}/2).$

Marginal distribution

If $(U_1,\dots ,U_r)\sim D_p(a_1,\dots ,a_{r+1})$, and if $s\le r$, then:

$(U_1,\dots ,U_s)\sim D_p(a_1,\dots ,a_s,\sum _{i=s+1}^{r+1}{a}_i)$

Conditional distribution

Also, with the same notation as above, the density of $(U_{s+1},\dots ,U_r)|(U_1,\dots ,U_s)$ is given by

$\frac{\prod _{i=s+1}^{r+1}\det {\left(U_i\right)}^{a_i-(p+1)/2}}{{\beta }_p(a_{s+1},\dots ,a_{r+1})\det {(I_p-\sum _{i=1}^s{U}_i)}^{\sum _{i=s+1}^{r+1}{a}_i-(p+1)/2}}$

where we write $U_{r+1}=I_p-\sum _{i=1}^r{U}_i$.

partitioned distribution

Suppose $(U_1,\dots ,U_r)\sim D_p(a_1,\dots ,a_{r+1})$ and suppose that $S_1,\dots ,S_t$ is a partition of $[r+1]=\{1,\dots r+1\}$ (that is, $\underset{i=1}{\overset{t}{\cup }}{S}_i=[r+1]$ and $S_i\cap S_j=\mathrm{\varnothing }$ if $i\ne j$). 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:

$(U_{(1)},\dots U_{(t)})\sim D_p(a_{(1)},\dots ,a_{(t)}).$

partitions

Suppose $(U_1,\dots ,U_r)\sim D_p(a_1,\dots ,a_{r+1})$. Define

$U_i=\left(\begin{array}{cc}{U}_{11(i)}& U_{12(i)}\\ U_{21(i)}& U_{22(i)}\end{array}\right)i=1,\dots ,r$

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

$(U_{11(1)},\dots ,U_{11(r)})\sim D_{p_1}(a_1,\dots ,a_{r+1})$

and

$(U_{22.1(1)},\dots ,U_{22.1(r)})\sim D_{p_2}(a_1-p_1/2,\dots ,a_r-p_1/2,a_{r+1}-p_1/2+p_{1}r/2).$

See also

• Inverse Dirichlet distribution

References

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