朴素稀有事件估计量的不确定性量化与置信区间
Uncertainty Quantification and Confidence Intervals for Naive Rare-Event Estimators
AI总结：
本文研究仅基于样本比例估计稀有事件概率的置信区间构造、覆盖有效性与紧致性，提出 Chernoff 和 Berry–Esseen 区间，并推广至命中数停止抽样情形。
摘要：
我们考虑使用朴素 Monte Carlo 输出的样本比例或收集到的数据来估计稀有事件概率。与采用方差缩减技术不同，这种朴素估计量没有先验的相对效率保证。另一方面，由于智能系统安全评估中近期涌现出复杂的稀有事件问题，具有效率保证的方差缩减可能面临实施挑战；这些挑战连同可用的计算或数据收集能力，促使人们使用这种朴素估计量。本文研究仅使用样本比例对稀有事件概率进行不确定性量化，即置信区间的构造、覆盖有效性和紧致性。除已知的正态区间、Wilson 区间和精确区间外，我们研究并比较了由 Chernoff 不等式和 Berry–Esseen 定理导出的两个新区间。此外，我们将结果推广到一种自然情形：抽样在达到目标稀有事件命中次数时停止。研究结果表明，正态区间和 Wilson 区间并不总是有效，但在半宽方面它们与新提出的有效区间接近。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。我们的新区间虽然比精确区间更保守，却为理解所考察区间的紧致性提供了有用洞见。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
