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\{\beta_p\left(a,b\right)\right\}^{-1} \det\left(U\right)^{a-(p+1)/2}\det\left(I_p-U\right)^{b-(p+1)/2}. $$

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



\beta_p\left(a,b\right)=\frac{\Gamma_p\left(a\right)\Gamma_p\left(b\right)}{\Gamma_p\left(a+b\right)} $$

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



\Gamma_p\left(a\right)= \pi^{p(p-1)/4}\prod_{i=1}^p\Gamma\left(a-(i-1)/2\right). $$

Distribution of matrix inverse
If $$U\sim B_p(a,b)$$ then the density of $$X=U^{-1}$$ is given by



\frac{1}{\beta_p\left(a,b\right)}\det(X)^{-(a+b)}\det\left(X-I_p\right)^{b-(p+1)/2} $$ provided that $$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{bmatrix}

U_{11} & U_{12} \\ U_{21} & U_{22} \end{bmatrix}$$

where $$U_{11}$$ is $$p_1\times p_1$$ and $$U_{22}$$ is $$p_2\times p_2$$, then defining the Schur complement $$U_{22\cdot 1}$$ as $$ U_{22}-U_{21}{U_{11}}^{-1}U_{12}$$ gives the following results:


 * $$U_{11}$$ is independent of $$U_{22\cdot 1}$$
 * $$U_{11}\sim B_{p_1}\left(a,b\right)$$
 * $$U_{22\cdot 1}\sim B_{p_2}\left(a-p_1/2,b\right)$$
 * $$U_{21}\mid U_{11},U_{22\cdot 1}$$ has an inverted matrix variate t distribution, specifically $$U_{21}\mid U_{11},U_{22\cdot 1}\sim IT_{p_2,p_1} \left(2b-p+1,0,I_{p_2}-U_{22\cdot 1},U_{11}(I_{p_1}-U_{11})\right).$$

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


 * $$U = S^{-1/2}S_1\left(S^{-1/2}\right)^T,$$

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 $$\Sigma$$.