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