In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a matrix gamma distribution is a generalization of the
gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
to
positive-definite matrices
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a c ...
.
[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. It is a more general version of the
Wishart distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.
It is a family of probability distributions define ...
, and is used similarly, e.g. as the
conjugate prior
In Bayesian probability theory, if the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions, and th ...
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
In statistics, the matrix ''t''-distribution (or matrix variate ''t''-distribution) is the generalization of the multivariate ''t''-distribution from vectors to matrices. The matrix ''t''-distribution shares the same relationship with the mult ...
.
This reduces to the
Wishart distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.
It is a family of probability distributions define ...
with
Notice that in this parametrization, the parameters
and
are not identified; the density depends on these two parameters through the product
.
See also
*
inverse matrix gamma distribution
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 conjugat ...
.
*
matrix normal distribution.
*
matrix t-distribution
In statistics, the matrix ''t''-distribution (or matrix variate ''t''-distribution) is the generalization of the multivariate ''t''-distribution from vectors to matrices. The matrix ''t''-distribution shares the same relationship with the multi ...
.
*
Wishart distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.
It is a family of probability distributions define ...
.
Notes
References
* Gupta, A. K.; Nagar, D. K. (1999) ''Matrix Variate Distributions'', Chapman and Hall/CRC
{{ProbDistributions, multivariate
Random matrices
Continuous distributions
Multivariate continuous distributions