2015/05/29 by Yun Yang, Martin J. Wainwright, Yang, Yun +3 · 3 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Methodology (stat.ME) #Statistical Methods and Inference #Statistics Theory (math.ST) #cs.LG #math.ST #stat.CO #stat.ME #stat.ML #stat.TH
paper · pdf · doi:10.48550/arxiv.1505.07925
42 pages, 3 figures
arxiv created 2015/05/29 · openalex publication_date 2015/05/29 · arxiv updated 2015/06/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee both variable-selection consistency and rapid mixing of a particular Metropolis-Hastings algorithm. The mixing time is linear in the number of covariates up to a logarithmic factor. Our proof controls the spectral gap of the Markov chain by constructing a canonical path ensemble that is inspired by the steps taken by greedy algorithms for variable selection.