Rare Event Simulation and Splitting for Discontinuous Random Variables

TL;DR

This paper addresses the challenge of estimating rare event probabilities when the underlying random variable has discontinuities, proposing three unbiased corrected estimators that adapt to both continuous and discontinuous cases.

Contribution

It introduces three new unbiased estimators for multilevel splitting methods that handle discontinuities in the distribution of the target variable, improving estimator robustness.

Findings

Estimators are unbiased regardless of discontinuities.

Efficiency demonstrated on a 2-D diffusive process.

Application to Boolean SAT problem shows practical effectiveness.

Abstract

Multilevel Splitting methods, also called Sequential Monte-Carlo or \emph{Subset Simulation}, are widely used methods for estimating extreme probabilities of the form $P[S(\mathbf{U}) > q]$ where $S$ is a deterministic real-valued function and $\mathbf{U}$ can be a random finite- or infinite-dimensional vector. Very often, $X := S(\mathbf{U})$ is supposed to be a continuous random variable and a lot of theoretical results on the statistical behaviour of the estimator are now derived with this hypothesis. However, as soon as some threshold effect appears in $S$ and/or $\mathbf{U}$ is discrete or mixed discrete/continuous this assumption does not hold any more and the estimator is not consistent. In this paper, we study the impact of discontinuities in the \emph{cdf} of $X$ and present three unbiased \emph{corrected} estimators to handle them. These estimators do not require to know in advance if $X$ is actually discontinuous or not and become all equal if $X$ is continuous. Especially, one of them has the same statistical properties in any case. Efficiency is shown on a 2-D diffusive process as well as on the \emph{Boolean SATisfiability problem} (SAT).