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Concept
Inverse-Wishart distribution
Summary
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 the inverse-Wishart distribution.
The probability density function of the inverse Wishart is:
where and are positive definite matrices, is the determinant, and Γp(·) is the multivariate gamma function.
If and is of size , then has an inverse Wishart distribution .
Suppose has an inverse Wishart distribution. Partition the matrices and conformably with each other
where and are matrices, then we have
is independent of and , where is the Schur complement of in ;
where is a matrix normal distribution;
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 , using the formula and the linear algebra identity :
(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.
Official source
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