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

TL;DR

This paper analyzes the asymptotic behavior of the multivariate spike-and-slab LASSO for variable and covariance selection, establishing posterior contraction rates and proposing de-biased confidence intervals with good coverage.

Contribution

It provides the first theoretical analysis of posterior contraction for the multivariate spike-and-slab LASSO and introduces de-biased intervals for covariate effects and residual correlations.

Findings

Posterior contraction rates are established for diverging p and q.

De-biased intervals achieve close-to-nominal frequentist coverage.

De-biased intervals for covariate effects are asymptotically valid.

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.