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Journal of Probability and Statistics
Volume 2014 (2014), Article ID 673657, 13 pages
Research Article

Bayesian Inference of a Multivariate Regression Model

1ZestFinance, 6636 Hollywood Boulevard, Los Angeles, CA 90028, USA
2Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA

Received 11 June 2014; Revised 14 October 2014; Accepted 28 October 2014; Published 24 November 2014

Academic Editor: Z. D. Bai

Copyright © 2014 Marick S. Sinay and John S. J. Hsu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We explore Bayesian inference of a multivariate linear regression model with use of a flexible prior for the covariance structure. The commonly adopted Bayesian setup involves the conjugate prior, multivariate normal distribution for the regression coefficients and inverse Wishart specification for the covariance matrix. Here we depart from this approach and propose a novel Bayesian estimator for the covariance. A multivariate normal prior for the unique elements of the matrix logarithm of the covariance matrix is considered. Such structure allows for a richer class of prior distributions for the covariance, with respect to strength of beliefs in prior location hyperparameters, as well as the added ability, to model potential correlation amongst the covariance structure. The posterior moments of all relevant parameters of interest are calculated based upon numerical results via a Markov chain Monte Carlo procedure. The Metropolis-Hastings-within-Gibbs algorithm is invoked to account for the construction of a proposal density that closely matches the shape of the target posterior distribution. As an application of the proposed technique, we investigate a multiple regression based upon the 1980 High School and Beyond Survey.