Matrix gamma distribution

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.^[1] 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.^[1]

Matrix gamma

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

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

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

See also

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