Inverse matrix gamma distribution

Inverse matrix gamma

Notation	[math]\displaystyle{ {\rm IMG}_{p}(\alpha,\beta,\boldsymbol\Psi) }[/math]
Parameters	[math]\displaystyle{ \alpha \gt (p - 1)/2 }[/math] shape parameter [math]\displaystyle{ \beta \gt 0 }[/math] scale parameter [math]\displaystyle{ \boldsymbol\Psi }[/math] scale (positive-definite real [math]\displaystyle{ p\times p }[/math] matrix)
Support	[math]\displaystyle{ \mathbf{X} }[/math] positive-definite real [math]\displaystyle{ p\times p }[/math] matrix
PDF	[math]\displaystyle{ \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) }[/math] [math]\displaystyle{ \Gamma_p }[/math] is the multivariate gamma function.

In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. 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.

This reduces to the inverse Wishart distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom when [math]\displaystyle{ \beta=2, \alpha=\frac{\nu}{2} }[/math].

See also

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