Inverse matrix gamma distribution

In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices.[1] 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	${\displaystyle {\rm {IMG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Psi }})}$
Parameters	${\displaystyle \alpha >(p-1)/2}$ shape parameter ${\displaystyle \beta >0}$ scale parameter ${\displaystyle {\boldsymbol {\Psi }}}$ scale (positive-definite real ${\displaystyle p\times p}$ matrix)
Support	${\displaystyle \mathbf {X} }$ positive-definite real ${\displaystyle p\times p}$ matrix
PDF	${\displaystyle {\frac {|{\boldsymbol {\Psi }}|^{\alpha }}{\beta ^{p\alpha }\Gamma _{p}(\alpha )}}|\mathbf {X} |^{-\alpha -(p+1)/2}\exp \left(-{\frac {1}{\beta }}{\rm {tr}}\left({\boldsymbol {\Psi }}\mathbf {X} ^{-1}\right)\right)}$ ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function.

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

See also

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

References

1. Iranmanesha, Anis; Arashib, M.; Tabatabaeya, S. M. M. (2010). "On Conditional Applications of Matrix Variate Normal Distribution". Iranian Journal of Mathematical Sciences and Informatics. 5 (2): 33–43.