  This paper introduces a new Markov Chain Monte Carlo method for Bayesian
variable selection in high dimensional settings. The algorithm is a
Hastings-Metropolis sampler with a proposal mechanism which combines a
Metropolis Adjusted Langevin (MALA) step to propose local moves associated with
a shrinkage-thresholding step allowing to propose new models. The geometric
ergodicity of this new trans-dimensional Markov Chain Monte Carlo sampler is
established. An extensive numerical experiment, on simulated and real data, is
presented to illustrate the performance of the proposed algorithm in comparison
with some more classical trans-dimensional algorithms.
