On the computational complexity of high-dimensional Bayesian variable selection

Yun Yang, Martin J. Wainwright, Michael I. Jordan

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)2497-2532
Number of pages36
JournalAnnals of Statistics
Volume44
Issue number6
DOIs
StatePublished - Dec 2016
Externally publishedYes

Keywords

  • Bayesian variable selection
  • High-dimensional inference
  • Markov chain
  • Rapid mixing
  • Spectral gap

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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