Matrix gamma distribution explained

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]

A matrix gamma distributions is identical to a Wishart distribution with

\beta \boldsymbol\Sigma = 2 V, \alpha = (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 .

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

Notes and References

1. 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.