2018/07/31 by Ray Bai, Malay Ghosh
Computer Science · Mathematics · #Algorithm #Applied mathematics #Bayes factor #Bayes' theorem #Bayesian Methods and Mixture Models #Bayesian inference #Bayesian linear regression #Bayesian probability #Beta distribution #Combinatorics #Computer science #False discovery rate #Hyperparameter #Marginal likelihood #Mathematical optimization #Mathematics #Minimax #Model selection #Monte Carlo method #Prime (order theory) #Prior probability #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics #stat.ME
paper · pdf · doi:10.5705/ss.202019.0037
37 pages, 4 figures, 3 tables. We have added a section on posterior computation and corrected the theoretical results. Sections on normal means estimation were removed in this updated technical report
arxiv created 2019/01/24 · openalex publication_date 2019/07/17 · arxiv updated 2019/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study high-dimensional Bayesian linear regression with a general beta prime distribution for the scale parameter. Under the assumption of sparsity, we show that appropriate selection of the hyperparameters in the beta prime prior leads to the (near) minimax posterior contraction rate when p ≫ n. For finite samples, we propose a data-adaptive method for estimating the hyperparameters based on marginal maximum likelihood (MML). This enables our prior to adapt to both sparse and dense settings, and under our proposed empirical Bayes procedure, the MML estimates are never at risk of collapsing to zero. We derive efficient Monte Carlo EM and variational EM algorithms for implementing our model, which are available in the R package NormalBetaPrime. Simulations and analysis of a gene expression data set illustrate our model's self-adaptivity to varying levels of sparsity and signal strengths.