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

9 September 2022
Yunyi Shen
Sameer K. Deshpande
ArXiv (abs)PDFHTML
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 qqq correlated responses are regressed onto ppp covariates. In this problem, the goal is to estimate a sparse matrix BBB of marginal covariate effects and a sparse precision matrix Ω\OmegaΩ, 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 BBB and on all the off-diagonal elements in the lower-triangle of Ω\OmegaΩ. Under mild assumptions, we establish the posterior contraction rate for the slightly modified mSSL posterior in the asymptotic regime where both ppp and qqq diverge with n.n.n. Our results imply that a slightly modified version of Deshpande et al.~(2019)'s mSSL procedure is asymptotically consistent.

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