期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2017
卷号：114
期号：19
页码：4925-4929
DOI：10.1073/pnas.1618780114
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：We study how much data a Bayesian observer needs to correctly infer the relative likelihoods of two events when both events are arbitrarily rare. Each period, either a blue die or a red die is tossed. The two dice land on side 1 with unknown probabilities p 1 and q 1 , which can be arbitrarily low. Given a data-generating process where p 1 ≥ c q 1 , we are interested in how much data are required to guarantee that with high probability the observer’s Bayesian posterior mean for p 1 exceeds ( 1 − δ ) c times that for q 1 . If the prior densities for the two dice are positive on the interior of the parameter space and behave like power functions at the boundary, then for every ϵ > 0 , there exists a finite N so that the observer obtains such an inference after n periods with probability at least 1 − ϵ whenever n p 1 ≥ N . The condition on n and p 1 is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.
关键词：rare event ; Bayes etimate ; uniform consistency ; multinomial distribution ; signaling game
