  A simple and efficient adaptive Markov Chain Monte Carlo (MCMC) method,
called the Metropolized Adaptive Subspace (MAdaSub) algorithm, is proposed for
sampling from high-dimensional posterior model distributions in Bayesian
variable selection. The MAdaSub algorithm is based on an independent
Metropolis-Hastings sampler, where the individual proposal probabilities of the
explanatory variables are updated after each iteration using a form of Bayesian
adaptive learning, in a way that they finally converge to the respective
covariates' posterior inclusion probabilities. We prove the ergodicity of the
algorithm and present a parallel version of MAdaSub with an adaptation scheme
for the proposal probabilities based on the combination of information from
multiple chains. The effectiveness of the algorithm is demonstrated via various
simulated and real data examples, including a high-dimensional problem with
more than 20,000 covariates.
