Bayesian posteriors for arbitrarily rare events.

Area of Science:

• Statistical inference
• Probability theory
• Bayesian analysis

Background:

• Understanding Bayesian inference with rare events is crucial.
• Determining data requirements for accurate likelihood comparisons is challenging.

Purpose of the Study:

• Quantify the amount of data needed for a Bayesian observer to reliably infer relative likelihoods of two arbitrarily rare events.
• Investigate the impact of prior density functions on data requirements.

Main Methods:

• Utilized a Bayesian framework with a data-generating process involving two dice (blue and red) with unknown, potentially very low, probabilities of landing on a specific side.
• Analyzed the observer's posterior mean under specific prior conditions (positive on the interior, power-like at the boundary).

Main Results:

• Established that for specific priors, a finite amount of data guarantees high-probability inference about relative likelihoods.
• Demonstrated that the derived condition on probabilities is optimal.
• Identified that the result may fail if prior densities decay exponentially at the boundary.

Conclusions:

• Finite data is sufficient for reliable Bayesian inference of rare event likelihoods under certain prior conditions.
• The behavior of prior densities at the boundary significantly impacts the required data quantity.
• The findings provide theoretical bounds for statistical learning with rare events.