Reference analysis of the signal + background model in counting experiments II. Approximate reference prior

TL;DR

This paper develops an approximate reference prior for Bayesian analysis of Poisson signal plus background models, showing it performs well across various background levels and simplifies computations compared to flat priors.

Contribution

It introduces a simple approximation to the reference prior for signal intensity in Poisson models, improving Bayesian inference accuracy and computational efficiency.

Findings

Approximate prior closely matches the true reference prior across background levels.

Limiting form of the prior is a Gamma distribution related to observed counts.

A 2-parameter fitting function effectively reproduces the reference prior for various backgrounds.

Abstract

The objective Bayesian treatment of a model representing two independent Poisson processes, labelled as "signal" and "background" and both contributing additively to the total number of counted events, is considered. It is shown that the reference prior for the parameter of interest (the signal intensity) can be well approximated by the widely (ab)used flat prior only when the expected background is very high. On the other hand, a very simple approximation (the limiting form of the reference prior for perfect prior background knowledge) can be safely used over a large portion of the background parameters space. The resulting approximate reference posterior is a Gamma density whose parameters are related to the observed counts. This limiting form is simpler than the result obtained with a flat prior, with the additional advantage of representing a much closer approximation to the reference posterior in all cases. Hence such limiting prior should be considered a better default or conventional prior than the uniform prior. On the computing side, it is shown that a 2-parameter fitting function is able to reproduce extremely well the reference prior for any background prior. Thus, it can be useful in applications requiring the evaluation of the reference prior for a very large number of times. [The published version JINST 9 (2014) T10006 has a typo in the normalization $N$ of eq.(2.6) that is fixed here.]