Matrix gamma distribution

Matrix gamma

Notation	${\mathrm{M}\mathrm{G}}_p(\alpha ,\beta ,\mathrm{\Sigma })$
Parameters	$\alpha >\frac{p-1}{2}$ shape parameter (real) $\beta >0$ scale parameter $\mathrm{\Sigma }$ scale (positive-definite real $p\times p$ matrix)
Support	$\mathbf{𝐗}$ positive-definite real $p\times p$ matrix
PDF	$\frac{|\mathrm{\Sigma }{|}^{-\alpha }}{{\beta }^{p\alpha }{\mathrm{\Gamma }}_p(\alpha )}|\mathbf{𝐗}{|}^{\alpha -\frac{p+1}{2}}\exp \left(\mathrm{t}\mathrm{r}(-\frac{1}{\beta }{\mathrm{\Sigma }}^{-1}\mathbf{𝐗})\right)$ ${\mathrm{\Gamma }}_p$ is the multivariate gamma function.

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. It is effectively a different parametrization 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.

A matrix gamma distributions is identical to a Wishart distribution with $\beta \mathrm{\Sigma }=2V,\alpha =\frac{n}{2}.$

Notice that the parameters $\beta$ and $\mathrm{\Sigma }$ are not identified; the density depends on these two parameters through the product $\beta \mathrm{\Sigma }$.

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