Matrix gamma distribution

Matrix gamma
Matrix gamma
Notation
Parameters	shape parameter (real); scale parameter; scale (positive-definite real matrix)
Support	positive-definite real matrix
PDF	is the multivariate gamma function.;

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]

Notation

{\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})

Parameters

$\alpha >{\frac {p-1}{2}}$ 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

${\frac {|{\boldsymbol {\Sigma }}|^{-\alpha }}{\beta ^{p\alpha }\,\Gamma _{p}(\alpha )}}|\mathbf {X} |^{\alpha -{\frac {p+1}{2}}}\exp \left({\rm {tr}}\left(-{\frac {1}{\beta }}{\boldsymbol {\Sigma }}^{-1}\mathbf {X} \right)\right)$

$\Gamma _{p}$ is the multivariate gamma function.

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

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

[1]

Matrix gamma distribution

See also

Notes

References

Related Articles

Matrix gamma
Notation	${\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})$
Parameters	$\alpha >{\frac {p-1}{2}}$ 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	${\frac {\|{\boldsymbol {\Sigma }}\|^{-\alpha }}{\beta ^{p\alpha }\,\Gamma _{p}(\alpha )}}\|\mathbf {X} \|^{\alpha -{\frac {p+1}{2}}}\exp \left({\rm {tr}}\left(-{\frac {1}{\beta }}{\boldsymbol {\Sigma }}^{-1}\mathbf {X} \right)\right)$ $\Gamma _{p}$ is the multivariate gamma function.