Bayesian correlation estimation

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.