On the posterior contraction of the multivariate spike-and-slab LASSO
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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 Ω, 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 Ω. 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.
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