A few brief notes on the equivalence of two expressions for statistical significance in point source detections

TL;DR

This paper demonstrates that two seemingly different formulas for assessing statistical significance in point source detection are mathematically equivalent, clarifying their relationship in Poisson-limited count maps.

Contribution

It proves the equivalence of two different expressions for p-values used in point source detection, bridging Bayesian and non-Bayesian approaches.

Findings

The two formulas are mathematically equivalent.

Clarifies the relationship between Bayesian and frequentist methods.

Provides a unified understanding of significance testing in this context.

Abstract

The problem of point source detection in Poisson-limited count maps has been addressed by two recent papers [M. Lampton, ApJ 436, 784 (1994); D. E. Alexandreas, et al., Nucl. Instr. Meth. Phys. Res. A 328, 570 (1993)]. Both papers consider the problem of determining whether there are significantly more counts in a source region than would be expected given the number of counts observed in a background region. The arguments in the two papers are quite different (one takes a Bayesian point of view and the other does not), and the suggested formulas for computing p-values appear to be different as well. It is shown here that the expressions provided by the authors of these two articles are in fact equivalent.