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Mixture models are one of the most widely used statistical tools when dealingwith data from heterogeneous populations. This paper considers thelong-standing debate over finite mixture and infinite mixtures and brings thetwo modelling strategies together, by showing that a finite mixture is simply arealization of a point process. Following a Bayesian nonparametric perspective,we introduce a new class of prior: the Normalized Independent Point Processes.We investigate the probabilistic properties of this new class. Moreover, wedesign a conditional algorithm for finite mixture models with a random numberof components overcoming the challenges associated with the Reversible Jumpscheme and the recently proposed marginal algorithms. We illustrate our modelon real data and discuss an important application in population genetics.

Is infinity that far? A Bayesian nonparametric perspective of finite mixture models