Sampling Conditionally on a Rare Event via Generalized Splitting

We propose and analyze a generalized splitting method to sample approximately from a distribution conditional on the occurrence of a rare event. This has important applications in a variety of contexts in operations research, engineering, and computational statistics. The method uses independent trials starting from a single particle. We exploit this independence to obtain asymptotic and nonasymptotic bounds on the total variation error of the sampler. Our main finding is that the approximation error depends crucially on the relative variability of the number of points produced by the splitting algorithm in one run and that this relative variability can be readily estimated via simulation. We illustrate the relevance of the proposed method on an application in which one needs to sample (approximately) from an intractable posterior density in Bayesian inference.