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.
follows an inverse Wishart distribution, denoted as
Important identities have been derived for the inverse-Wishart distribution.
[2] The probability density function of the inverse Wishart is:[3] where
positive definite matrices,
is the multivariate gamma function.
has an inverse Wishart distribution
has an inverse Wishart distribution.
matrices, then we have Suppose we wish to make inference about a covariance matrix
are independent p-variate Gaussian variables drawn from a
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
and the linear algebra identity
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.
(1982) "Applied Multivariate Analysis", 2nd ed.
(Dover Publications, New York), after reparameterizing the degree of freedom to be consistent with the p.d.f.
, which simplifies to: The covariance of elements of
are given by: The same results are expressed in Kronecker product form by von Rosen[6] as follows: where There appears to be a typo in the paper whereby the coefficient of
, and that the expression for the mean square inverse Wishart, corollary 3.1, should read To show how the interacting terms become sparse when the covariance is diagonal, let
denotes the matrix vectorization operator.
Then the second moment matrix becomes which is non-zero only when involving the correlations of diagonal elements of
, all other elements are mutually uncorrelated, though not necessarily statistically independent.
The variances of the Wishart product are also obtained by Cook et al.[7] in the singular case and, by extension, to the full rank case.
, one degree of freedom being relinquished by estimation of the sample mean in the latter.
the marginal distribution of the leading diagonal element is thus and by rotating
end-around a similar result applies to all diagonal elements
A corresponding result in the complex Wishart case was shown by Brennan and Reed[9] and the uncorrelated inverse complex Wishart
was shown by Shaman[10] to have diagonal statistical structure in which the leading diagonal elements are correlated, while all other element are uncorrelated.