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:

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

Matrix variate beta distribution
Notation	${\rm {B}}_{p}(a,b)$
Parameters	$a,b$
Support	$p\times p$ matrices with both $U$ and $I_{p}-U$ positive definite
PDF	$\left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.$

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

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

Theorems

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 $a,b>(p-1)/2$.

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 $\Sigma$.

See also

• Matrix variate Dirichlet distribution

References

• A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.
• S. K. Mitra 1970. "A density-free approach to matrix variate beta distribution". The Indian Journal of Statistics, Series A, (1961-2002), volume 32, number 1 (March 1970), pp81-88.