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We study the asymptotic properties of Deshpande et al.\ (2019)'s multivariatespike-and-slab LASSO (mSSL) procedure for simultaneous variable and covarianceselection in the sparse multivariate linear regression problem. In thatproblem, $q$ correlated responses are regressed onto $p$ covariates and themSSL works by placing separate spike-and-slab priors on the entries in thematrix of marginal covariate effects and off-diagonal elements in the uppertriangle of the residual precision matrix. Under mild assumptions about thesematrices, we establish the posterior contraction rate for the mSSL posterior inthe asymptotic regime where both $p$ and $q$ diverge with $n.$ By``de-biasing'' the corresponding MAP estimates, we obtain confidence intervalsfor each covariate effect and residual partial correlation. In extensivesimulation studies, these intervals displayed close-to-nominal frequentistcoverage in finite sample settings but tended to be substantially longer thanthose obtained using a version of the Bayesian bootstrap that randomlyre-weights the prior. We further show that the de-biased intervals forindividual covariate effects are asymptotically valid.

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