Bayesian variable selection and the Swendsen-Wang algorithm
David Nott & Peter J. Green
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.
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