Matrix Variate Beta Distribution

In statistics, the matrix variate beta distribution is a generalization of the beta distribution. If $U$ is a $p\times p$ positive definite matrix with a matrix variate beta distribution, and $a,b>(p-1)/2$ are real parameters, we write $U\sim B_p\left(a,b\right)$ (sometimes $B_p^I\left(a,b\right)$). The probability density function for $U$ is:

:$\left\^ \det\left(U\right)^\det\left(I_p-U\right)^.$

Here $\beta_p\left(a,b\right)$ is the multivariate beta function:

:$\beta_p\left(a,b\right)=\frac$

where $\Gamma_p\left(a\right)$ is the multivariate gamma function given by

:$\Gamma_p\left(a\right)= \pi^\prod_^p\Gamma\left(a-(i-1)/2\right).$

Theorems

Distribution of matrix inverse

If $U\sim B_p(a,b)$ then the density of $X=U^$ is given by

:$\frac\det(X)^\det\left(X-I_p\right)^$
provided that $X>I_p$ and $a,b>(p-1)/2$.

Orthogonal transform

If $U\sim B_p(a,b)$ and $H$ is a constant $p\times p$ orthogonal matrix, then $HUH^T\sim B(a,b).$

Also, if $H$ is a random orthogonal $p\times p$ matrix which is independent of $U$, then $HUH^T\sim B_p(a,b)$, distributed independently of $H$.

If $A$ is any constant $q\times p$, $q\leq p$ matrix of rank $q$, then $AUA^T$ has a generalized matrix variate beta distribution, specifically $AUA^T\sim GB_q\left(a,b;AA^T,0\right)$.

Partitioned matrix results

If $U\sim B_p\left(a,b\right)$ and we partition $U$ as

:$U=\begin U_ & U_ \\ U_ & U_ \end$

where $U_$ is $p_1\times p_1$ and $U_$ is $p_2\times p_2$, then defining the Schur complement $U_$ as $U_-U_^U_$ gives the following results:

* $U_$ is independent of $U_$
* $U_\sim B_\left(a,b\right)$
* $U_\sim B_\left(a-p_1/2,b\right)$
* $U_\mid U_,U_$ has an inverted matrix variate t distribution, specifically $U_\mid U_,U_\sim IT_ \left(2b-p+1,0,I_-U_,U_(I_-U_)\right).$

Wishart results

Mitra proves the following theorem which illustrates a useful property of the matrix variate beta distribution. Suppose $S_1,S_2$ are independent Wishart $p\times p$ matrices $S_1\sim W_p(n_1,\Sigma), S_2\sim W_p(n_2,\Sigma)$. Assume that $\Sigma$ is positive definite and that $n_1+n_2\geq p$. If

:$U = S^S_1\left(S^\right)^T,$

where $S=S_1+S_2$, then $U$ has a matrix variate beta distribution $B_p(n_1/2,n_2/2)$. In particular, $U$ is independent of $\Sigma$.

See also

* Matrix variate Dirichlet distribution

References

*
*{{cite journal , first=S. K. , last=Mitra , year=1970 , title=A density-free approach to matrix variate beta distribution , journal=The Indian Journal of Statistics , series=Series A (1961–2002) , volume=32 , issue=1 , pages=81–88 , jstor=25049638

Random matrices
Multivariate continuous distributions