Conclusions

In this paper we have dealt with the often debated issue of `ratios of small numbers of events', approaching it from a probabilistic perspective. After having shown the difference between predicting numbers of counts (and their ratios) and inferring the Poisson parameters (and their ratios) on the base of the observed numbers of counts, the attention has been put on the latter, “a problem in the probability of causes, ...  the essential problem of the experimental method” [21]. Having the paper a didactic intent, the basic ideas of probabilistic inference have been reminded, together with the use of conjugate priors in order to get closed results with minimum effort. It has been also shown how to perform the so called `propagation of uncertainties' in closed forms, which has required, for the purposes of this work, to derive the probability density function of the ratio of Gamma distributed variables. And, as byproducts, the `curious' pdf of the ratio of two uniform variables has been derived and a new derivation of the formula to get the pdf of a function of variables has been devised.

The importance of graphical models has been stressed. In fact, they are not only very useful to form a global, clearer vision of the problem, but also to possibly take into account alternative models. In the case of rather simple models it has been shown how to write down the joint distribution of all variables, from which the pdf of the variables of interest follows. In some cases, thanks to reasonable (or at least well stated) assumptions, closed results have been obtained, but we have also seen how to use tools based on MCMC, both to check the closed results and to tackle more realistic models (samples of programming code are provided in Appendix B).

Finally, as far as the issue of `combination of ratios' is concerned, it has been shown how the solution depends crucially on the physical model describing the variation of the rates and/or their ratio in function of an external variable. Therefore only general indications on how to approach the problem have been given, highly recommending the use of MCMC tools (my preference for small problem with limited amount of data goes presently to JAGS/rjags, but particle physicists might prefer BAT [45], or perhaps the more recent, Julia [46] based, BAT.jl [47]).

I am indebted to Alfredo (Dino) Esposito for many discussions on the probabilistic and technical aspects the paper, some of which admittedly based on Ref. [1], and for valuable comments on the manuscript.