On the On-Off Problem: An Objective Bayesian Analysis

TL;DR

This paper introduces an objective Bayesian method for the On-Off problem, enabling accurate inference of signal significance and strength in low-count scenarios common in astrophysics, surpassing traditional frequentist approaches.

Contribution

It provides a universal Bayesian solution for the On-Off problem that works for any count number, including zero, and unifies inference of significance, strength, and limits.

Findings

Method is valid for any count number, including zero.

Applied successfully to Gamma Ray Burst data.

Offers a unified Bayesian framework for signal inference.

Abstract

The On-Off problem, aka. Li-Ma problem, is a statistical problem where a measured rate is the sum of two parts. The first is due to a signal and the second due to a background, both of which are unknown. Mostly frequentist solutions are being used that are only adequate for high count numbers. When the events are rare such an approximation is not good enough. Indeed, in high-energy astrophysics this is often the rule rather than the exception. I will present a universal objective Bayesian solution that depends only on the initial three parameters of the On-Off problem: the number of events in the "on" region, the number of events in the "off" region, and their ratio-of-exposure. With a two-step approach it is possible to infer the signal's significance, strength, uncertainty or upper limit in a unified a way. The approach is valid without restrictions for any count number including zero and may be widely applied in particle physics, cosmic-ray physics and high-energy astrophysics. I apply the method to Gamma Ray Burst data.