Zero-Class Poisson for Rare-Event Studies

TL;DR

This paper introduces a Bayesian statistical framework for zero-class Poisson models, crucial for analyzing rare events across various scientific fields, and compares different priors to optimize estimation accuracy.

Contribution

It develops a Bayesian theory for zero-class Poisson models, evaluates priors, and demonstrates how zero-inflated and Negative-binomial models relate to Poisson under limited information.

Findings

Maximum-entropy prior yields minimal bias and risk.

Bayesian methods outperform classical solutions for ZCD.

Zero-inflated and Negative-binomial models reduce to Poisson with limited data.

Abstract

We developed a statistical theory of zero-count-detector (ZCD), which is defined as a zero-class Poisson under conditions outlined in the paper. ZCD is often encountered in the studies of rare events in physics, health physics, and many other fields where counting of events occurs. We found no acceptable solution to ZCD in classical statistics and affirmed the need for the Bayesian statistics. Several uniform and reference priors were studied and we derived Bayesian posteriors, point estimates, and upper limits. It was showed that the maximum-entropy prior, containing the most information, resulted in the smallest bias and the lowest risk, making it the most admissible and acceptable among the priors studied. We also investigated application of zero-inflated Poisson and Negative-binomial distributions to ZCD. It was showed using Bayesian marginalization that, under limited information, these distributions reduce to the Poisson distribution.