Probabilistic Bounded Relative Error Property for Learning Rare Event Simulation Techniques

In rare event simulation, we look for estimators such that the relative accuracy of the output is ''controlled'' when the rarity is getting more and more critical. Different robustness properties have been defined in the literature, that an estimator is expected to satisfy. Though, those properties are not adapted to estimators for which the estimators come from a parametric family and the optimal parameter is learned and random. For this reason, we motivate in this paper the need to define probabilistic robustness properties, because the accuracy of the resulting estimator is therefore random. We especially focus on the so-called probabilistic bounded relative error property. We additionally provide sufficient conditions, both in general and Markov settings, to satisfy such a property, illustrate them and simple but standard examples, and hope that it will foster discussions and new works in the area.