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.

翻译：我们研究了Deshpande等人(2019)提出的多元尖峰-厚板LASSO(mSSL)方法在稀疏多元线性回归问题中同时进行变量选择和协方差选择的渐近性质。在该问题中，$q$个相关响应变量被回归到$p$个协变量上，mSSL方法通过对边际协变量效应矩阵中的元素以及残差精度矩阵上三角部分的非对角元素分别施加尖峰-厚板先验来实现选择功能。在对这些矩阵的温和假设下，我们建立了在$p$和$q$均随$n$发散的渐近框架下mSSL后验分布的收缩速率。通过对相应的最大后验概率(MAP)估计进行"去偏"处理，我们获得了每个协变量效应和残差偏相关系数的置信区间。在大量模拟研究中，这些区间在有限样本设定下展现出接近名义水平的频率覆盖概率，但其长度往往显著长于通过随机重加权先验的贝叶斯自举法所获得的区间。我们进一步证明，针对个体协变量效应的去偏置信区间具有渐近有效性。