Bayesian variable selection and the Swendsen-Wang algorithm

The need to explore model uncertainty in linear regression models with many predictors has motivated improvements in Markov chain Monte Carlo sampling algorithms for Bayesian variable selection. Traditional sampling algorithms for Bayesian variable selection may perform poorly when there are severe multicollinearities amongst the predictors. In this paper we describe a new sampling method based on an analogy with the Swendsen-Wang algorithm for the Ising model, and which can give substantial improvements over traditional sampling schemes in the presence of multicollinearity. In linear regression with a given set of potential predictors we can index different possible models by a binary parameter vector which indicates which of the predictors are included or excluded. By thinking of the posterior distribution of this parameter as a binary spatial field, we can approximate the posterior distribution by an Ising model and then apply a modified Swendsen-Wang algorithm for sampling from the posterior where dependence among parameters is reduced by conditioning on auxiliary variables. Performance of the method is described for both simulated and real data.