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. Currently used sampling algorithms for Bayesian variable selection may per-form poorly when there are severe multicollinearities among the predictors. This article describes a new sampling method based on an analogy with the Swendsen-Wang algorithm for the Ising model, and which can give substantial improvements over alternative sampling schemes in the presence of multicollinearity. In linear regression with a given set of potential predictors we can index possible models by a binary parameter vector that 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 use auxiliary variable methods inspired by the Swendsen-Wang algorithm for the Ising model to sample 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.