Inverse matrix gamma distribution

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

In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices.[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.[citation needed]

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

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.