Inverse matrix gamma distribution

Inverse matrix gamma

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

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

This reduces to the inverse Wishart distribution with $\beta=2, \alpha=\frac{n}{2}$.

See also

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

References

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