On the computational complexity of high-dimensional Bayesian variable selection
Yun Yang, Martin J. Wainwright, Michael I. Jordan
The Annals of Statistics · 2016 · Volume 44 · Issue 6
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.
Keywords
60J10, 62F15, Bayesian variable selection, high-dimensional inference, Markov chain, rapid mixing, spectral gap
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