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December 2016 On the computational complexity of high-dimensional Bayesian variable selection
Yun Yang, Martin J. Wainwright, Michael I. Jordan
Ann. Statist. 44(6): 2497-2532 (December 2016). DOI: 10.1214/15-AOS1417

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.

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Yun Yang. Martin J. Wainwright. Michael I. Jordan. "On the computational complexity of high-dimensional Bayesian variable selection." Ann. Statist. 44 (6) 2497 - 2532, December 2016. https://doi.org/10.1214/15-AOS1417

Information

Received: 1 May 2015; Revised: 1 September 2015; Published: December 2016
First available in Project Euclid: 23 November 2016

zbMATH: 1359.62088
MathSciNet: MR3576552
Digital Object Identifier: 10.1214/15-AOS1417

Subjects:
Primary: 62F15
Secondary: 60J10

Keywords: Bayesian variable selection , high-dimensional inference , Markov chain , rapid mixing , spectral gap

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 6 • December 2016
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