TY - JOUR
T1 - Bayesian posteriors for arbitrarily rare events
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 4925
LP - 4929
DO - 10.1073/pnas.1618780114
VL - 114
IS - 19
AU - Fudenberg, Drew
AU - He, Kevin
AU - Imhof, Lorens A.
Y1 - 2017/05/09
UR - http://www.pnas.org/content/114/19/4925.abstract
N2 - Many decision problems in contexts ranging from drug safety tests to game-theoretic learning models require Bayesian comparisons between the likelihoods of two events. When both events are arbitrarily rare, a large data set is needed to reach the correct decision with high probability. The best result in previous work requires the data size to grow so quickly with rarity that the expectation of the number of observations of the rare event explodes. We show for a large class of priors that it is enough that this expectation exceeds a prior-dependent constant. However, without some restrictions on the prior the result fails, and our condition on the data size is the weakest possible.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 p1 and q1, which can be arbitrarily low. Given a data-generating process where p1≥cq1, we are interested in how much data are required to guarantee that with high probability the observer’s Bayesian posterior mean for p1 exceeds (1−δ)c times that for q1. 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 np1≥N. The condition on n and p1 is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.
ER -