Notation M G p ( α , β , Σ ) {\displaystyle {\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})} Parameters shape parameter (real) β > 0 {\displaystyle \beta >0} scale parameter Σ {\displaystyle {\boldsymbol {\Sigma }}} scale (positive-definite real p × p {\displaystyle p\times p} matrix) Support X {\displaystyle \mathbf {X} } positive-definite real p × p {\displaystyle p\times p} matrix PDF | Σ | − α β p α Γ p ( α ) | X | α − ( p + 1 ) / 2 exp ( t r ( − 1 β Σ − 1 X ) ) {\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)} Γ p {\displaystyle \Gamma _{p}} is the multivariate gamma function. | ||
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.
This reduces to the Wishart distribution with
References
Matrix gamma distribution Wikipedia(Text) CC BY-SA