Matrix variate beta distribution explained

In statistics, the matrix variate beta distribution is a generalization of the beta distribution. If

U is a p x 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 ^{I\left(a,b\right)} _{p} ). The probability density function for U is:\left\{\beta ^{-1} _{p\left(a,b\right)\right\}} \det\left(U\right) ^{a-(p+1)/2} \det\left(I ^{b - (p +1 )/ 2} _{p - U\right)} .

Here

\beta _{p\left(a,b\right)} is the multivariate beta function:\beta _{p\left(a,b\right) = (\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 ^{p\Gamma\left(a - (i -1 )/ 2 \right).} _{i =1}

Theorems

Distribution of matrix inverse

If

U\sim B _{p(a,b)} then the density of X = U ^{-1} is given by(1)/(\beta _{p\left(a,b\right)}) \det(X) ^{-(a+b)} \det\left(X - I ^{b - (p +1 )/ 2} _{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 x p orthogonal matrix, then HUH ^{T\sim} B(a,b).

Also, if

H is a random orthogonal p x 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 x p , q\leq p matrix of rank q , then AUA ^{T} has a generalized matrix variate beta distribution, specifically AUA ^{T\sim} GB ^{T, 0 \right)} _{q\left(a,b;AA} .

Partitioned matrix results

If

U\sim B _{p\left(a,b\right)} and we partition U asU = \begin{bmatrix} U _{11} & U _{12} \\ U _{21} & U _{22} \end{bmatrix}

where

U _{11} is p _{1 x} p _{1} and U _{22} is p _{2 x} p _{2} , then defining the Schur complement U _{22 ⋅} as U _{22} - U _{21} {U _{11} }^U_ gives the following results:U _{11} is independent of U _{22 ⋅} U _{11} \sim B _{p _{1}} \left(a,b\right) U _{22 ⋅} \sim B _{p _{2}} \left(a - p _{1/2,b\right)} U _{21} \mid U _{11} ,U _{22 ⋅} has an inverted matrix variate t distribution, specifically U _{21} \mid U _{11} ,U _{22 ⋅} \sim IT _{p _{2,p} _{1}} \left( 2 b - p +1 , 0 ,I _{p _{2}} - U _{22 ⋅} ,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 x 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 . IfU = S ^{-1/2} S ^{-1 / 2} _{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

• Book: Gupta, A. K. . D. K. . Nagar . 1999 . Matrix Variate Distributions . Chapman and Hall . 1-58488-046-5 .
• S. K. . Mitra . 1970 . A density-free approach to matrix variate beta distribution . The Indian Journal of Statistics . Series A (1961–2002) . 32 . 1 . 81–88 . 25049638 .