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[subscript 1] and q[subscript 1], which can be arbitrarily low. Given a data-generating process where p[subscript 1] ≥cq[subscript 1], we are interested in how much data are required to guarantee that with high probability the observer's Bayesian posterior mean for p[subscript 1] exceeds (1-δ)c times that for q[subscript 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 np 1 ≥N. The condition on n and p[subscript 1] is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.
Author: Fudenberg, Drew; He, Kevin; Imhof, Lorens A.
Department: Massachusetts Institute of Technology. Department of Economics
Publisher: National Academy of Sciences
Date Issued: 2017-03
ISSN: 0027-84241091-6490
Citation: Fudenberg, Drew, Kevin He, and Lorens A. Imhof. “Bayesian Posteriors for Arbitrarily Rare Events.” Proceedings of the National Academy of Sciences 114, no. 19 (April 25, 2017): 4925–4929. © 2018 National Academy of Sciences
Version: Final published version
Terms of Use: Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use
Journal: Proceedings of the National Academy of Sciences
No comments:
Post a Comment