We consider the long-standing problem of Bayesian variable selection in a linear regression model. Variable selection is a complicated task in high dimensional settings where the number of regression parameters P is much larger than the number of observations N. In this context, it is crucial to introduce sparsity assumptions based on the prior knowledge that only a few number of regression parameters are significant. Using a sequence of observations from a linear regression model, the aims are (i) to determine which components of the regression vector are active and explain the observations and (ii) to estimate the regression vector.

In this work, we introduce a new MCMC algorithm, called Shrinkage-Thresholding MALA (STMALA), designed to sample sparse regression vectors by jointly sampling a model and a regression vector in this model. This algorithm, which is a transdimensional MCMC method, relies on MALA (see [Roberts and Tweedie, 1996]). The proposal distribution of MALA is based on the computation of the gradient of the logarithm of the target distribution. In order to both deal with a non-differentiable target posterior distribution and to actually set some components to zero, we propose to combine MALA with a shrinkage-thresholding operator:

- – we first compute a noisy gradient step involving the term of the logarithm of the target distribution which is continuously differentiable;
- then a shrinkage-thresholding operator is applied to ensure sparsity and shrink small values of the regression parameters toward zero.

Such an algorithm is motivated by Bayesian variable selection with non-smooth priors. This algorithm can perform global moves from one model to a rather distant other one, which allows to explore efficiently high dimensional spaces in comparison to local move algorithms, like reversible jump MCMC (RJMCMC – see [Green 1995]). The geometric ergodicity of this new algorithm is proved for a large class of target distributions.

Joint work with Gersende Fort, Sylvain Le Corff and Éric Moulines.

References:

– P. Green, Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, *Biometrika* 82(4) 1995.

– G.O. Roberts, R.L. Tweedie, Exponential convergence of Langevin distributions and their discrete approximations, *Bernouilli* 2(4) 1996.

Keywords: Markov Chain Monte Carlo, Proximal operators, Bayesian variable selection.