Notation I M G p ( α , β , Ψ ) {\displaystyle {\rm {IMG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Psi }})} Parameters α {\displaystyle \alpha } shape parameter (real) β > 0 {\displaystyle \beta >0} scale parameter Ψ {\displaystyle {\boldsymbol {\Psi }}} 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 | Ψ | α β p α Γ p ( α ) | X | − α − ( p + 1 ) / 2 exp ( t r ( − 1 β Ψ X − 1 ) ) {\displaystyle {\frac {|{\boldsymbol {\Psi }}|^{\alpha }}{\beta ^{p\alpha }\Gamma _{p}(\alpha )}}|\mathbf {X} |^{-\alpha -(p+1)/2}\exp \left({\rm {tr}}\left(-{\frac {1}{\beta }}{\boldsymbol {\Psi }}\mathbf {X} ^{-1}\right)\right)} Γ p {\displaystyle \Gamma _{p}} is the multivariate gamma function. | ||
In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. It is a more general version of the inverse Wishart distribution, and is used similarly, e.g. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution. The compound distribution resulting from compounding a matrix normal with an inverse matrix gamma prior over the covariance matrix is a generalized matrix t-distribution.
This reduces to the inverse Wishart distribution with
References
Inverse matrix gamma distribution Wikipedia(Text) CC BY-SA