- Inverse-Wishart distribution
statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability density functiondefined on matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normaldistribution.
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
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. [Cite book
Kanti V. Mardia, J. T. Kent and J. M. Bibby
title = Multivariate Analysis
year = 1979
isbn = 0-12-471250-9]
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 complementof in ;
iii) , where is a
matrix normal distribution;
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 is times the sample covariance matrix.
Because the prior and posterior distributions are the same family, we say the inverse Wishart distribution is conjugate to the multivariate Gaussian.
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::
univariatespecialization of the inverse-Wishart distribution is the inverse-gamma distribution. With (i.e. univariate) and , and the probability density functionof the inverse-Wishart distribution becomes
i.e., the inverse-gamma distribution, where is the ordinary
A generalization is the
Matrix normal distribution
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