On the Error of Naive Rare-Event Monte Carlo Estimator

We consider the estimation of rare-event probabilities using sample proportions output by naive Monte Carlo. Unlike using variance reduction techniques, this naive estimator does not have a priori relative efficiency guarantee. On the other hand, due to the recent surge of sophisticated rare-event problems arising in safety evaluations of intelligent systems, efficiency-guaranteed variance reduction may face implementation challenges, which motivate one to look at naive estimators. In this paper we investigate this naive rare-event estimator, particularly its conservativeness level and the guarantees in using it to construct confidence bounds for the target probability. We show that the half-width of a valid confidence interval is typically scaled proportional to the magnitude of the target probability and inverse square-root with the number of positive outcomes in the Monte Carlo. We also derive and compare several valid confidence bounds constructed from various techniques.