论文标题
多变量尖峰和斜纹拉索的后收缩和不确定性定量
Posterior contraction and uncertainty quantification for the multivariate spike-and-slab LASSO
论文作者
论文摘要
我们研究了Deshpande等人（2019年）的多元尖峰和slab lasso（MSSL）程序的渐近性能，以同时可变和协方差选择在稀疏的多元线性回归问题中。在这个问题中，$ q $相关的响应会回归到$ p $协变量中，而MSSL可以通过在边缘协变量效果的矩阵中放置单独的尖峰和单杆先验，并在残差精确矩阵的上部三角形的矩阵中。在对这些矩阵的温和假设下，我们在渐近方案中建立了MSSL后部的后验收缩率，在$ p $和$ q $ diverge中，$n。$ by $n。$ by“ de de-de-de-de-de-de-de-de-de-de-de-de sibiasing''相应的地图估计值，我们获得了每个协方差效应和残留部分相关的置信区间。在广泛的仿真研究中，这些间隔在有限的样本设置中显示出接近新神经的频繁覆盖范围，但往往比使用使用贝叶斯引导版本获得的贝叶斯自举的版本更长的时间，该版本随机重新重量。我们进一步表明，单个协变量效应的偏差间隔渐近有效。
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.
