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)

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

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.