  Mixture models are one of the most widely used statistical tools when dealing
with data from heterogeneous populations. This paper considers the
long-standing debate over finite mixture and infinite mixtures and brings the
two modelling strategies together, by showing that a finite mixture is simply a
realization 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, we
design a conditional algorithm for finite mixture models with a random number
of components overcoming the challenges associated with the Reversible Jump
scheme and the recently proposed marginal algorithms. We illustrate our model
on real data and discuss an important application in population genetics.
