Dimension‐free mixing for high‐dimensional Bayesian variable selection

Yang et al. proved that the symmetric random walk Metropolis–Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high‐dimensional assumptions. We propose a novel Markov chain Monte Carlo (MCMC) sampler using an informed proposal scheme, which we prove achieves a much faster mixing time that is independent of the number of covariates, under the assumptions of Yang et al. To the best of our knowledge, this is the first high‐dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called ‘two‐stage drift condition’ to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high‐dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis.