Inverse matrix gamma distribution

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

In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices.^[1] It is a more general version of the inverse Wishart distribution, and is used similarly, e.g. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution. The compound distribution resulting from compounding a matrix normal with an inverse matrix gamma prior over the covariance matrix is a generalized matrix t-distribution.^{[citation needed]}

Notation

{\rm {IMG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Psi }})

Parameters

$\alpha >(p-1)/2$ shape parameter
$\beta >0$ scale parameter

{\boldsymbol {\Psi }}

scale (positive-definite real

p\times p

matrix)

Support

\mathbf {X}

positive-definite real

p\times p

matrix

PDF

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

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

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This reduces to the inverse Wishart distribution with $\nu$ degrees of freedom when $\beta =2,\alpha ={\frac {\nu }{2}}$ .

[1]

Inverse matrix gamma distribution

See also

References

Related Articles

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