Bayesian Linear Regression for Multivariate Responses Under Group Sparsity

We study the frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. Two features of the model are unique: (i) group sparsity is imposed on the predictors. (ii) the covariance matrix is unknown and its dimensions can be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a Wishart prior with increasing dimension on the inverse of the covariance matrix. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving a $\ell_2/\ell_1$-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein-von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory through constructing a suitable test from the first principle by bounding moments of likelihood ratio statistics around points in the alternative. This leads to the posterior concentrates around the truth with respect to the average log-affinity. The technique of obtaining the posterior contraction rate could be useful in many other problems.