Methods for Constructing Priors for Bayesian Covariance Matrix Estimation

• Activity Number: 211
• Type: Invited
• Date/Time: Tuesday, August 9, 2005 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Christopher K. Carter*+
• Companies: CSIRO
• Keywords: Constrained Wishart distribution ; Gaussian copula model ; Gaussian graphical model ; Multivariate analysis ; Multivariate probit model ; Partial correlations

Abstract

In this paper, we present a Bayesian approach for obtaining parsimony in the covariance matrix of Gaussian data. We give methods for constructing priors for the covariance matrix that allow the offdiagonal elements of the concentration matrix to be zero. The priors have normalizing constants for each possible model size, rather than for each possible model, which gives a tractable number of normalizing constants that need to be estimated. We show how these normalizing constants can be estimated using MCMC methods, and apply our methods to two types of prior for the concentration matrix. The first type is a mixture of constrained Wisharts. The second decomposes the concentration matrix into a function of partial correlations and conditional variances or marginal variances and uses a mixture distribution on the matrix of partial correlations. We also show how to extend the methodology to construct priors that allow the offdiagonal elements of a covariance matrix, a correlation matrix, or the inverse of a correlation matrix to be zero.