Posterior moments of scalar functions of a covariance matrix

Abstract

Given multivariate normal data and a certain spherically invariant prior distribution on the covariance matrix, it is desired to estimate the moments of the posterior marginal distributions of some scalar functions of the covariance matrix by importance sampling. To this end a family of distributions is defined on the group of orthogonal matrices and a procedure is proposed for selecting one of these distributions for use as a weighting distribution in the importance sampling process. In an example estimates are calculated for the posterior mean and variance of each element in the covariance matrix expressed in the original coordinates, for the posterior mean of each element in the correlation matrix expressed in the original coordinates, and for the posterior mean of each element in the covariance matrix expressed in the coordinates of the principal variables.