Bayesian posteriors for arbitrarily rare events
Name
Bayesian posteriors for arbitrarily rare events.pdf
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3.38 MB
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Author(s)
Fudenberg, Drew
He, Kevin
Imhof, Lorens A.
Date Issued
March 2017
Journal
Proceedings of the National Academy of Sciences
Publisher
National Academy of Sciences
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
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Economics
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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.
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DOI of Published Version
https://doi.org/10.1073/pnas.1618780114
