Bayesian posteriors for arbitrarily rare events.

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 [Formula: see text] with unknown probabilities [Formula: see text] and [Formula: see text], which can be arbitrarily low. Given a data-generating process where [Formula: see text], we are interested in how much data are required to guarantee that with high probability the observer's Bayesian posterior mean for [Formula: see text] exceeds [Formula: see text] times that for [Formula: see text] 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 [Formula: see text] there exists a finite [Formula: see text] so that the observer obtains such an inference after [Formula: see text] periods with probability at least [Formula: see text] whenever [Formula: see text] The condition on [Formula: see text] and [Formula: see text] is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.