# Matrix gamma distribution

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.[1] 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.[1]

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

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