Split Sampling: Expectations, Normalisation and Rare Events

TL;DR

This paper introduces split sampling, a novel Monte Carlo methodology for estimating high-dimensional expectations and rare event probabilities, demonstrated through network path and Gaussian mixture applications.

Contribution

The paper develops split sampling, a new auxiliary variable MCMC-based approach for rare event probability estimation, with theoretical derivation and practical implementations.

Findings

Accurate estimation of rare event probabilities in high dimensions.

Comparison shows split sampling outperforms cross entropy and nested sampling.

Method implemented in R package SplitSampling.

Abstract

In this paper we develop a methodology that we call split sampling methods to estimate high dimensional expectations and rare event probabilities. Split sampling uses an auxiliary variable MCMC simulation and expresses the expectation of interest as an integrated set of rare event probabilities. We 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 network 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. We discuss the relationship between our method and other alternatives such as the product of conditional probability estimator and importance sampling. The methods developed here are available in the R package: SplitSampling.