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 ${\displaystyle U_{1},\ldots ,U_{r}}$ are ${\displaystyle p\times p}$ positive definite matrices with ${\displaystyle I_{p}-\sum _{i=1}^{r}U_{i}}$ also positive-definite, where ${\displaystyle I_{p}}$ is the ${\displaystyle p\times p}$ identity matrix. Then we say that the ${\displaystyle U_{i}}$ have a matrix variate Dirichlet distribution, ${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)}$, if their joint probability density function is

${\displaystyle \left\{\beta _{p}\left(a_{1},\ldots ,a_{r},a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r}\det \left(U_{i}\right)^{a_{i}-(p+1)/2}\det \left(I_{p}-\sum _{i=1}^{r}U_{i}\right)^{a_{r+1}-(p+1)/2}}$

where ${\displaystyle a_{i}>(p-1)/2,i=1,\ldots ,r+1}$ and ${\displaystyle \beta _{p}\left(\cdots \right)}$ is the multivariate beta function.

If we write ${\displaystyle U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{i}}$ then the PDF takes the simpler form

${\displaystyle \left\{\beta _{p}\left(a_{1},\ldots ,a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r+1}\det \left(U_{i}\right)^{a_{i}-(p+1)/2},}$

on the understanding that ${\displaystyle \sum _{i=1}^{r+1}U_{i}=I_{p}}$.

Theorems

generalization of chi square-Dirichlet result

Suppose ${\displaystyle S_{i}\sim W_{p}\left(n_{i},\Sigma \right),i=1,\ldots ,r+1}$ are independently distributed Wishart ${\displaystyle p\times p}$ positive definite matrices. Then, defining ${\displaystyle U_{i}=S^{-1/2}S_{i}\left(S^{-1/2}\right)^{T}}$ (where ${\displaystyle S=\sum _{i=1}^{r+1}S_{i}}$ is the sum of the matrices and ${\displaystyle S^{1/2}\left(S^{-1/2}\right)^{T}}$ is any reasonable factorization of ${\displaystyle S}$), we have

${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(n_{1}/2,...,n_{r+1}/2\right).}$

Marginal distribution

If ${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r+1}\right)}$, and if ${\displaystyle s\leq r}$, then:

${\displaystyle \left(U_{1},\ldots ,U_{s}\right)\sim D_{p}\left(a_{1},\ldots ,a_{s},\sum _{i=s+1}^{r+1}a_{i}\right)}$

Conditional distribution

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

${\displaystyle {\frac {\prod _{i=s+1}^{r+1}\det \left(U_{i}\right)^{a_{i}-(p+1)/2}}{\beta _{p}\left(a_{s+1},\ldots ,a_{r+1}\right)\det \left(I_{p}-\sum _{i=1}^{s}U_{i}\right)^{\sum _{i=s+1}^{r+1}a_{i}-(p+1)/2}}}}$

where we write ${\displaystyle U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{i}}$.

partitioned distribution

Suppose ${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r+1}\right)}$ and suppose that ${\displaystyle S_{1},\ldots ,S_{t}}$ is a partition of ${\displaystyle \left[r+1\right]=\left\{1,\ldots r+1\right\}}$ (that is, ${\displaystyle \cup _{i=1}^{t}S_{i}=\left[r+1\right]}$ and ${\displaystyle S_{i}\cap S_{j}=\emptyset }$ if ${\displaystyle i\neq j}$). Then, writing ${\displaystyle U_{(j)}=\sum _{i\in S_{j}}U_{i}}$ and ${\displaystyle a_{(j)}=\sum _{i\in S_{j}}a_{i}}$ (with ${\displaystyle U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{r}}$), we have:

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

partitions

Suppose ${\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r+1}\right)}$. Define

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

where ${\displaystyle U_{11(i)}}$ is ${\displaystyle p_{1}\times p_{1}}$ and ${\displaystyle U_{22(i)}}$ is ${\displaystyle p_{2}\times p_{2}}$. Writing the Schur complement ${\displaystyle U_{22\cdot 1(i)}=U_{21(i)}U_{11(i)}^{-1}U_{12(i)}}$ we have

${\displaystyle \left(U_{11(1)},\ldots ,U_{11(r)}\right)\sim D_{p_{1}}\left(a_{1},\ldots ,a_{r+1}\right)}$

and

${\displaystyle \left(U_{22.1(1)},\ldots ,U_{22.1(r)}\right)\sim D_{p_{2}}\left(a_{1}-p_{1}/2,\ldots ,a_{r}-p_{1}/2,a_{r+1}-p_{1}/2+p_{1}r/2\right).}$

See also

• Inverse Dirichlet distribution

References

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