Matrix gamma distribution

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

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. It is 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 $\beta 𝜮=2V,\alpha =\frac{n}{2}.$

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

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