Matrix variate beta distribution

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

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

Here ${\displaystyle {\beta }_p(a,b)}$ is the multivariate beta function:

${\displaystyle {\beta }_p(a,b)=\frac{{\mathrm{\Gamma }}_p\left(a\right){\mathrm{\Gamma }}_p\left(b\right)}{{\mathrm{\Gamma }}_p(a+b)}}$

where ${\displaystyle {\mathrm{\Gamma }}_p\left(a\right)}$ is the multivariate gamma function given by

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

Theorems

Distribution of matrix inverse

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

${\displaystyle \frac{1}{{\beta }_p(a,b)}det(X{)}^{-(a+b)}det{(X-I_p)}^{b-(p+1)/2}}$

provided that ${\displaystyle X>I_p}$ and ${\displaystyle a,b>(p-1)/2}$.

Orthogonal transform

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

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

If ${\displaystyle A}$ is any constant ${\displaystyle q\times p}$, ${\displaystyle q\le p}$ matrix of rank ${\displaystyle q}$, then ${\displaystyle AU{A}^T}$ has a generalized matrix variate beta distribution, specifically ${\displaystyle AU{A}^T\sim G{B}_q(a,b;A{A}^T,0)}$.

Partitioned matrix results

If ${\displaystyle U\sim B_p(a,b)}$ and we partition ${\displaystyle U}$ as

${\displaystyle U=\left[\begin{array}{cc}{U}_{11}& U_{12}\\ U_{21}& U_{22}\end{array}\right]}$

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

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

Wishart results

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

${\displaystyle U=S^{-1/2}{S}_1{\left(S^{-1/2}\right)}^T,}$

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

See also

• Matrix variate Dirichlet distribution

References

• Gupta, A. K.; Nagar, D. K. (1999). Matrix Variate Distributions. Chapman and Hall. ISBN 1-58488-046-5. 
• Mitra, S. K. (1970). "A density-free approach to matrix variate beta distribution". The Indian Journal of Statistics. Series A (1961–2002) 32 (1): 81–88. 

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