We consider the estimation of rare-event probabilities using sample proportions output by naive Monte Carlo or collected data. 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, coupled with the availability of computation or data collection power, motivate the use of such a naive estimator. In this paper we study the uncertainty quantification, namely the construction, coverage validity and tightness of confidence intervals, for rare-event probabilities using only sample proportions. In addition to the known normality, Wilson's and exact intervals, we investigate and compare them with two new intervals derived from Chernoff's inequality and the Berry-Esseen theorem. Moreover, we generalize our results to the natural situation where sampling stops by reaching a target number of rare-event hits. Our findings show that the normality and Wilson's intervals are not always valid, but they are close to the newly developed valid intervals in terms of half-width. In contrast, the exact interval is conservative, but safely guarantees the attainment of the nominal confidence level. Our new intervals, while being more conservative than the exact interval, provide useful insights in understanding the tightness of the considered intervals.

翻译：本文考虑使用朴素蒙特卡洛方法或收集数据得到的样本比例估计稀有事件概率。与使用方差缩减技术不同，该朴素估计器不具备先验的相对效率保证。然而，由于近年来智能系统安全性评估中涌现出的复杂稀有事件问题，具有效率保证的方差缩减方法可能面临实施挑战，加之计算或数据收集能力的普及，推动了这类朴素估计器的使用。本文研究仅利用样本比例对稀有事件概率进行不确定度量化，即置信区间的构建、覆盖有效性及紧致性。除已知的正态区间、威尔逊区间和精确区间外，我们考察并比较了基于切尔诺夫不等式和贝里-埃森定理推导出的两种新区间。此外，我们将结果推广到通过达到目标稀有事件命中数停止抽样的自然情形。研究结果表明，正态区间和威尔逊区间并非始终有效，但它们的半宽与新发展出的有效区间相近。相比之下，精确区间较为保守，但能安全保证达到名义置信水平。我们提出的新区间虽比精确区间更保守，但为理解所考虑区间的紧致性提供了有益见解。