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In statistics, the inverse matrix gamma distribution is a generalization of the
inverse gamma distribution In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
to positive-definite matrices. It is a more general version of the
inverse Wishart distribution In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the c ...
, 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 t ...
of the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements o ...
of a
multivariate normal distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
or
matrix normal distribution In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. Definition The probability density ...
. The
compound distribution In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some ...
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 In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the c ...
with \nu degrees of freedom when \beta=2, \alpha=\frac.


See also

*
inverse Wishart distribution In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the c ...
. * matrix gamma distribution. *
matrix normal distribution In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. Definition The probability density ...
. * matrix t-distribution. *
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 defin ...
.


References

Random matrices Continuous distributions Multivariate continuous distributions {{matrix-stub