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In rare-event simulation, an importance sampling (IS) estimator is regardedas efficient if its relative error, namely the ratio between its standarddeviation and mean, is sufficiently controlled. It is widely known that when arare-event set contains multiple "important regions" encoded by the so-calleddominating points, IS needs to account for all of them via mixing to achieveefficiency. We argue that in typical experiments, missing less significantdominating points may not necessarily cause inefficiency, and the traditionalanalysis recipe could suffer from intrinsic looseness by using relative error,or in turn estimation variance, as an efficiency criterion. We propose a newefficiency notion, which we call probabilistic efficiency, to tighten this gap.In particular, we show that under the standard Gartner-Ellis large deviationsregime, an IS that uses only the most significant dominating points issufficient to attain this efficiency notion. Our finding is especially relevantin high-dimensional settings where the computational effort to locate alldominating points is enormous.

Over-Conservativeness of Variance-Based Efficiency Criteria and Probabilistic Efficiency in Rare-Event Simulation