Matrix gamma distribution

Matrix gamma

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

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. It is a more general version of the Wishart distribution, and is used similarly, e.g. as the conjugate prior of the precision matrix of a multivariate normal distribution and matrix normal distribution. The compound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is a generalized matrix t-distribution.

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

See also

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

References

• Gupta, A. K.; Nagar, D. K. (1999) Matrix Variate Distributions, Chapman and Hall/CRC ISBN 978-1584880462