|| degrees of freedom (real)
, scale matrix (pos. def)
|| is positive definite
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 covariance matrix of a multivariate normal distribution.
We say follows an inverse Wishart distribution, denoted as , if its inverse has a Wishart distribution . Important identities have been derived for Inverse-Wishart distribution.
The probability density function of the inverse Wishart is:
where and are positive definite matrices, and Γp(·) is the multivariate gamma function.
Distribution of the inverse of a Wishart-distributed matrix
If and is of size , then has an inverse Wishart distribution .
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 is the Schur complement of in ;
iii) , where is a matrix normal distribution;
iv) , where ;
Suppose we wish to make inference about a covariance matrix whose prior has a distribution. If the observations are independent p-variate Gaussian variables drawn from a distribution, then the conditional distribution has a distribution, where .
Because the prior and posterior distributions are the same family, we say the inverse Wishart distribution is conjugate to the multivariate Gaussian.
Due to its conjugacy to the multivariate Gaussian, it is possible to marginalize out (integrate out) the Gaussian's parameter .
(this is useful because the variance matrix is not known in practice, but because is known a priori, and can be obtained from the data, the right hand side can be evaluated directly). The inverse-Wishart distribution as a prior can be constructed via existing transferred prior knowledge.
The following is based on Press, S. J. (1982) "Applied Multivariate Analysis", 2nd ed. (Dover Publications, New York), after reparameterizing the degree of freedom to be consistent with the p.d.f. definition above.
The variance of each element of :
The variance of the diagonal uses the same formula as above with , which simplifies to:
The covariance of elements of are given by:
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 inverse multivariate gamma distribution.
Another generalization has been termed the generalized inverse Wishart distribution, . A positive definite matrix is said to be distributed as if is distributed as . Here denotes the symmetric matrix square root of , the parameters are positive definite matrices, and the parameter is a positive scalar larger than . Note that when is equal to an identity matrix, . This generalized inverse Wishart distribution has been applied to estimating the distributions of multivariate autoregressive processes.
A different type of generalization is the normal-inverse-Wishart distribution, essentially the product of a multivariate normal distribution with an inverse Wishart distribution.