Inverse matrix gamma distribution

Inverse matrix gamma
Notation	${\mathrm{I}\mathrm{M}\mathrm{G}}_p(\alpha ,\beta ,𝜳)$
Parameters	$\alpha >(p-1)/2$ shape parameter $\beta >0$ scale parameter $𝜳$ scale (positive-definite real $p\times p$ matrix)
Support	$𝐗$ positive-definite real $p\times p$ matrix
PDF	$\frac{|𝜳{|}^{\alpha }}{{\beta }^{p\alpha }{\Gamma }_p(\alpha )}|𝐗{|}^{-\alpha -(p+1)/2}\exp (-\frac{1}{\beta }\mathrm{t}\mathrm{r}\left(𝜳{𝐗}^{-1}\right))$ ${\Gamma }_p$ 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 $\nu$ 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.