Matrix gamma distribution

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 $$\beta \boldsymbol\Sigma = 2 V, \alpha=\frac{n}{2}.$$

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