Some results on the number of coincidences under exchangeability

One of the most renowned probability problems is the birthday problem: if n balls are randomly dropped into k boxes, what is the chance of a match, that is, that two or more balls fall in the same box? The classical answer is given under the assumption that balls are dropped independently and uniformly into each box. If k=365, the answer gives the probability of two or more coincident birthdays, in a group of n individuals. Some results on the birthday problem with non-uniform occurrence probabilities can be found in the literature. Recently, the birthday problem has been considered in a Bayesian framework. Exact calculations, for the chance of a match, have been presented under uniform prior and symmetric Dirichlet prior. Moreover a Poisson approximation for the law of the number of coincidences has been proved, under general Dirichlet priors. This last result relies on the Chen-Stein approximation methods, applied to negatively associated random variables. In the present paper, the approximation problem for the number of matches is faced, under a general prior, that is only assuming exchangeability. We provide necessary and sufficient conditions for the law of the number of matches to be well approximated by a mixture of Poisson distributions. Moreover, we characterize the prior distribution and give sufficient conditions involving observable quantities alone. This enables, in a Bayesian setting, to decide whether to assume a Poisson model, for the number of matches, on the base of prior information.