Bayesian sparse seemingly unrelated regressions model with variable selection and covariance estimation via the horseshoe

초록

We consider a general Bayesian sparse seemingly unrelated regressions (SSUR) model, where the number of predictors and the dimension of multivariate responses are relatively large compared to the sample size. We propose a new Bayesian approach to the SSUR model, called the HS+GHS+ using an elementwise horseshoe+ prior on the regression coefficients and a graphical horseshoe+ prior to the off-diagonal elements of the precision matrix, respectively, for sparsity. With the inverse-gamma mixture representation of half-Cauchy distribution and columnwise sampling scheme for the precision matrix derived from Schur complement, a computationally efficient Markov chain Monte Carlo (MCMC) algorithm and its fast alternative Variational Bayes (VB) algorithm are presented. We provide a theoretical justification for the proposed method through posterior consistency and improved Kullback-Leibler (KL) risk bound over existing methods. Finally, we illustrate our method with simulation studies and two real datasets.

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