Ratio of counts vs ratio of rates in Poisson processes

TL;DR

This paper clarifies the distinction between predicting event counts and inferring underlying rates in Poisson processes, providing analytical formulas, graphical models, and Monte Carlo methods for practical inference including real-world complications.

Contribution

It introduces a probabilistic framework with graphical models for analyzing ratios of counts and rates in Poisson processes, including methods for real-world data complexities.

Findings

Closed-form expressions for rates and ratios under assumptions.

Monte Carlo methods for ratio inference and validation.

Guidelines for incorporating efficiencies and systematics.

Abstract

The often debated issue of `ratios of small numbers of events' is approached from a probabilistic perspective, making a clear distinction between the predictive problem (forecasting numbers of events we might count under well stated assumptions, and therefore of their ratios) and inferential problem (learning about the relevant parameters of the related probability distribution, in the light of the observed number of events). The quantities of interests and their relations are visualized in a graphical model (`Bayesian network'), very useful to understand how to approach the problem following the rules of probability theory. In this paper, written with didactic intent, we discuss in detail the basic ideas, however giving some hints of how real life complications, like (uncertain) efficiencies and possible background and systematics, can be included in the analysis, as well as the possibility that the ratio of rates might depend on some physical quantity. The simple models considered in this paper allow to obtain, under reasonable assumptions, closed expressions for the rates and their ratios. Monte Carlo methods are also used, both to cross check the exact results and to evaluate by sampling the ratios of counts in the cases in which large number approximation does not hold. In particular it is shown how to make approximate inferences using a Markov Chain Monte Carlo using JAGS/rjags. Some examples of R and JAGS code are provided.