Dimension-Free Mixing for High-Dimensional Bayesian Variable Selection

Abstract 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 much faster time independent of number covariates, assumptions To best our knowledge, this first result rigorously shows rate MCMC methods can be fast enough to offset computational cost local posterior evaluation. Motivated by theoretical analysis sampler, further new approach called ‘two-stage drift condition’ studying convergence rates chains on general state spaces, useful obtaining tight complexity bounds in settings. The practical advantages are illustrated both simulation studies and real data analysis.