Split Sampling: Expectations, Normalisation and Rare Events

In this paper we develop a methodology that we call split sampling methods which estimate expectations in high dimensions more precisely than other available methods. Split sampling methods are attractive for computing normalisation constants and estimating rare event probabilities. We implement our method using an auxiliary variable MCMC simulation with the expectation of interest expressed as an integrated set of rare event probabilities and derive our estimator from a Rao-Blackwellised estimate of a marginal auxiliary variable distribution. We illustrate our method with two applications. First, we compute a shortest path rare event probability and compare our method to estimation to a cross entropy approach. Then, we compute a normalisation constant of a high dimensional mixture of Gaussians and compare our estimate to one based on nested sampling. Finally, we discuss the relationship between our method and alternatives such as bridge sampling and the Wang-Landau algorithm. The methods developed here are available in the R package: SplitSampling.