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 \mathbf{X} follows an inverse Wishart distribution, denoted as, if its inverse has a Wishart distribution. Important identities have been derived for the inverse-Wishart distribution.

Density

The probability density function of the inverse Wishart is: where \mathbf{X} and are p\times p positive definite matrices, | \cdot | is the determinant, and is the multivariate gamma function.

Theorems

Distribution of the inverse of a Wishart-distributed matrix

If and is of size p \times p, 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 {\mathbf A} and conformably with each other where and are matrices, then we have

Conjugate 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. 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 {\mathbf A} 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.

Moments

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. Let with \nu \ge p and, so that. The mean: The variance of each element of \mathbf{X}: The variance of the diagonal uses the same formula as above with i=j, which simplifies to: The covariance of elements of \mathbf{X} are given by: The same results are expressed in Kronecker product form by von Rosen as follows: where There appears to be a typo in the paper whereby the coefficient of is given as c_1 rather than c_2, 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 and introduce some arbitrary parameters u, v, w: where denotes the matrix vectorization operator. Then the second moment matrix becomes which is non-zero only when involving the correlations of diagonal elements of W^{-1}, all other elements are mutually uncorrelated, though not necessarily statistically independent. The variances of the Wishart product are also obtained by Cook et al. in the singular case and, by extension, to the full rank case. Muirhead shows in Theorem 3.2.8 that if is distributed as and V is an arbitrary vector, independent of A then and , one degree of freedom being relinquished by estimation of the sample mean in the latter. Similarly, Bodnar et.al. further find that and setting the marginal distribution of the leading diagonal element is thus and by rotating V end-around a similar result applies to all diagonal elements. A corresponding result in the complex Wishart case was shown by Brennan and Reed and the uncorrelated inverse complex Wishart was shown by Shaman to have diagonal statistical structure in which the leading diagonal elements are correlated, while all other element are uncorrelated.

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