Inverse matrix gamma distribution

Inverse matrix gamma

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