Bayesian correlation estimation

• Published in: Biometrika Vol. 91; no. 1; pp. 1 - 14
• Main Authors: Liechty, John C., Liechty, Merrill W., Müller, Peter
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01-03-2004; Biometrika Trust, University College London; Oxford University Press for Biometrika Trust; Oxford Publishing Limited (England)
• Series: Biometrika
• Subjects: Applications; Biology, psychology, social sciences; Correlations; Covariance matrices; Covariance matrix; Eigenvalues; Eigenvectors; Exact sciences and technology; Finance companies; Inference; Markov chains; Mathematics; Mixture prior; Modeling; Multivariate analysis; Oil companies; Parametric inference; Parametric models; Probabilities; Probability and statistics; Sciences and techniques of general use; Separation strategy; Simulation; Simulations; Standard deviation; Statistics; Variables; Biometrics; Bayes estimation; Mixed distribution; Prior information; Stock return; Cluster analysis (statistics); Correlation matrix; Shrinkage estimator; Statistical estimation; Parametric model; Covariance matrix; Prior probability; Population genetics; Parametric method; Statistical method; Separation method; Correlation analysis; Positive definite matrix; Posterior distribution
• ISSN: 0006-3444; 1464-3510
• DOI: 10.1093/biomet/91.1.1

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.