Inverse matrix gamma distribution

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.

Inverse matrix gamma

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

This reduces to the inverse Wishart distribution with $\nu$ degrees of freedom when $\beta =2,\alpha ={\frac {\nu }{2}}$.

See also

• inverse Wishart distribution.
• matrix gamma distribution.
• matrix normal distribution.
• matrix t-distribution.
• Wishart distribution.