Bayesian posteriors for arbitrarily rare events
Author(s)Fudenberg, Drew; He, Kevin; Imhof, Lorens A.
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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.
DepartmentMassachusetts Institute of Technology. Department of Economics
Proceedings of the National Academy of Sciences
National Academy of Sciences
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
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