# On the posterior contraction of the multivariate spike-and-slab LASSO

** Abstract ** : We study the asymptotic properties of the multivariate spike-and-slab LASSO (mSSL) proposed by Deshpande et al.(2019) for simultaneous variable and covariance selection. Specifically, we consider the sparse multivariate linear regression problem where q correlated responses are regressed onto p covariates. In this problem, the goal is to estimate a sparse matrix $B$ of marginal covariate effects and a sparse precision matrix $\Omega$, which captures the residual conditional dependence structure of the outcomes. The mSSL works by placing continuous spike and slab priors on all the entries of $B$ and on all the off-diagonal elements in the lower-triangle of $\Omega$. Under mild assumptions, we establish the posterior contraction rate for the slightly modified mSSL posterior in the asymptotic regime where both $p$ and $q$ diverge with $n.$ Our results imply that a slightly modified version of Deshpande et al. (2019)’s mSSL procedure is asymptotically consistent.

A pre-print version of the paper is available at arXiv:2207.07020.

**Recommended citation**: Shen, Y., and Deshpande, S.K. (2022+). "On the posterior contraction of the multivariate spike-and-slab LASSO." *arXiv preprint* arXiv:2209.04389