Matrix gamma distribution

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

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

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

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

This reduces to the Wishart distribution with \beta=2, \alpha=\frac{n}{2}.

See also[edit]


  1. ^ a b Iranmanesh, Anis, M. Arashib and S. M. M. Tabatabaey (2010). "On Conditional Applications of Matrix Variate Normal Distribution". Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43.


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