In
statistics, the inverse matrix gamma distribution is a generalization of the
inverse gamma distribution 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 ...
.
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 co ...
, 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
covariance matrix 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 som ...
resulting from compounding a matrix normal with an inverse matrix gamma prior over the covariance 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 multi ...
.
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 co ...
with
degrees of freedom when
.
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 co ...
.
*
matrix gamma distribution
In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.Iranmanesh, Anis, M. Arashib and S. M. M. Tabatabaey (2010)"On Conditional Applications of Matrix Variate Normal 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
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 defi ...
.
References
Random matrices
Continuous distributions
Multivariate continuous distributions
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