Bayesian Posteriors For Arbitrarily Rare Events

TL;DR

This paper analyzes the data requirements for Bayesian inference to reliably distinguish between two arbitrarily rare events, establishing conditions under which accurate posterior estimates are achievable.

Contribution

It provides theoretical bounds on the data needed for Bayesian observers to correctly infer relative likelihoods of rare events, considering different prior behaviors.

Findings

Bayesian posterior mean exceeds a threshold with high probability after sufficient data

Conditions on prior densities affect the success of inference for rare events

The derived data requirements are optimal under specified prior assumptions

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 $p_1$ and $q_1$, which can be arbitrarily low. Given a data-generating process where $p_1\ge c q_1$, we are interested in how much data is required to guarantee that with high probability the observer's Bayesian posterior mean for $p_1$ exceeds $(1-\delta)c$ times that for $q_1$. 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 $\epsilon>0,$ there exists a finite $N$ so that the observer obtains such an inference after $n$ periods with probability at least $1-\epsilon$ whenever $np_1\ge N$. The condition on $n$ and $p_1$ is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.