© 2004 by Biometrika Trust
Bayesian correlation estimation
1 Departments of Marketing and Statistics, Pennsylvania State University, University Park, Pennsylvania 16802-3007, U.S.A.jcl12{at}psu.edu 2 LeBow College of Business, Drexel University, Philadelphia, Pennsylvania 19104–2875, U.S.A.merrill{at}drexel.edu 3 Department of Biostatistics, University of Texas M.D. Anderson Cancer Center, Houston, Texas 77030-4009, U.S.A.pm{at}mdacc.tmc.edu
We propose prior probability models for variance-covariance matrices in order to address two important issues.First, the models allow a researcher to represent substantive prior information about the strength of correlations among a set of variables. Secondly, even in the absence of such information, the increased flexibility of the models mitigates dependence on strict parametric assumptions in standard prior models. For example, the model allows a posteriori different levels of uncertainty about correlations among different subsets of variables. We achieve this by including a clustering mechanism in the prior probability model. Clustering is with respect to variables and pairs of variables. Our approach leads to shrinkage towards a mixture structure implied by the clustering. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalising constants that are functions of parameters of interest. The normalising constants result from the restriction that the correlation matrix be positive definite. We discuss examples based on simulated data, a stock return dataset and a population genetics dataset.
Key Words: Covariance matrix; Mixture prior; Separation strategy
Received August 2002. Revised March 2003