Posterior contraction and uncertainty quantification for the multivariate spike-and-slab LASSO

Abstract: We study the asymptotic properties of Deshpande et al.\ (2019)'s multivariate spike-and-slab LASSO (mSSL) procedure for simultaneous variable and covariance selection in the sparse multivariate linear regression problem. In that problem, $q$ correlated responses are regressed onto $p$ covariates and the mSSL works by placing separate spike-and-slab priors on the entries in the matrix of marginal covariate effects and off-diagonal elements in the upper triangle of the residual precision matrix. Under mild assumptions about these matrices, we establish the posterior contraction rate for the mSSL posterior in the asymptotic regime where both $p$ and $q$ diverge with $n.$ By ``de-biasing'' the corresponding MAP estimates, we obtain confidence intervals for each covariate effect and residual partial correlation. In extensive simulation studies, these intervals displayed close-to-nominal frequentist coverage in finite sample settings but tended to be substantially longer than those obtained using a version of the Bayesian bootstrap that randomly re-weights the prior. We further show that the de-biased intervals for individual covariate effects are asymptotically valid.