A Bayesian model for measurements by counting

Counting processes occur very often in several scientific and technological problems. The concept of numerousness and, consequently, the counting of a number of items are at the base of many high-level measurements, as well as in everyday life applications. The occurrence of counting errors is a real issue that needs to be addressed. It might occur, for example, that one fails in counting one or more objects because of some reasons, such as human or instrumental errors. In such a case, the measurand, i.e., the number of items intended to be counted, is underestimated. On the other hand, one may count non-existing objects, hence obtaining an overestimate of the measurand. In a previous paper, a general model for measurements by counting was proposed which allows an uncertainty evaluation consistent with the general framework of the “Guide to the expression of uncertainty in measurement”. The present work considers the same scenario but facing the problem from a Bayesian point of view. In particular, we propose a (discrete) likelihood function of the counted objects, modelling the counting errors, and derive the posterior probability mass function for some selected prior distributions. Incorporating prior information on the measurand and the available knowledge on the counting errors, the proposed Bayesian model is able to provide measurand estimates corrected for such errors and accompanied by appropriate measurement uncertainties.