# Matrix gamma distribution

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. It is a more general version of the Wishart distribution, and is used similarly, e.g. as the conjugate prior of the precision matrix of a multivariate normal distribution and matrix normal distribution. The compound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is a generalized matrix t-distribution.

Notation ${\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})$ $\alpha >0$ shape parameter (real) $\beta >0$ scale parameter ${\boldsymbol {\Sigma }}$ scale (positive-definite real $p\times p$ matrix) $\mathbf {X}$ positive-definite real $p\times p$ matrix ${\frac {|{\boldsymbol {\Sigma }}|^{-\alpha }}{\beta ^{p\alpha }\Gamma _{p}(\alpha )}}|\mathbf {X} |^{\alpha -(p+1)/2}\exp \left({\rm {tr}}\left(-{\frac {1}{\beta }}{\boldsymbol {\Sigma }}^{-1}\mathbf {X} \right)\right)$ $\Gamma _{p}$ is the multivariate gamma function.

This reduces to the Wishart distribution with $\beta =2,\alpha ={\frac {n}{2}}.$ 