Matrix gamma distribution

Matrix gamma

Notation	[math]\displaystyle{ {\rm MG}_{p}(\alpha,\beta,\boldsymbol\Sigma) }[/math]
Parameters	[math]\displaystyle{ \alpha \gt \frac{p-1}{2} }[/math] shape parameter (real) [math]\displaystyle{ \beta \gt 0 }[/math] scale parameter [math]\displaystyle{ \boldsymbol\Sigma }[/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\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) }[/math] [math]\displaystyle{ \Gamma_p }[/math] is the multivariate gamma function.

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 [math]\displaystyle{ \beta \boldsymbol\Sigma = 2 V, \alpha=\frac{n}{2}. }[/math]

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

See also

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

References

• Gupta, A. K.; Nagar, D. K. (1999) Matrix Variate Distributions, Chapman and Hall/CRC ISBN:978-1584880462