In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability density function defined on matrices. In Bayesian statistics it is used as the conjugate for the covariance matrix of a
multivariate normal distribution.
We say follows an inverse Wishart distribution, denoted as , if its probability density function is written as follows:
where is a matrix. The matrix is assumed to be positive definite.
Distribution of the inverse of a Wishart-distributed matrix
If and is , then has an inverse Wishart distribution with probability density function:
where and is the multivariate gamma function.
Marginal and conditional distributions from an inverse Wishart-distributed matrix
Suppose has an inverse Wishart distribution. Partition the matrices and conformably with each other
where and are matrices, then we have
i) is independent of and , where ;
iii) , where is a matrix normal distribution;
If and has the a priori distribution then the conditional distribution of is .
If , then
A univariate specialization of the inverse-Wishart distribution is the inverse-gamma distribution. With (i.e. univariate) and , and the probability density function of the inverse-Wishart distribution becomes
i.e., the inverse-gamma distribution, where is the ordinary Gamma function.
A generalization is the normal-inverse-Wishart distribution.