BAYESIAN LINEAR REGRESSION WITH SPARSE PRIORS

• 成果类型: Article
• 署名作者: Castillo, Ismael; Schmidt-Hieber, Johannes; Van der Vaart, Aad
• 署名单位: Sorbonne Universite; Universite Paris Cite; Leiden University - Excl LUMC; Leiden University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1334
• 发表日期: 2015
• 页码: 1986-2018
• 关键词: variable-selection; stochastic search; convergence rate; model selection; Optimal Rates; posterior; Lasso; needles; straw

摘要

We study full Bayesian procedures for high-dimensional linear regression under sparsity constraints. The prior is a mixture of point masses at zero and continuous distributions. Under compatibility conditions on the design matrix, the posterior distribution is shown to contract at the optimal rate for recovery of the unknown sparse vector, and to give optimal prediction of the response vector. It is also shown to select the correct sparse model, or at least the coefficients that are significantly different from zero. The asymptotic shape of the posterior distribution is characterized and employed to the construction and study of credible sets for uncertainty quantification.