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

Mixture models are one of the most widely used statistical tools when dealing with data from heterogeneous populations. Following a Bayesian nonparametric perspective, we introduce a new class of priors: the Normalized Independent Point Process. We investigate the probabilistic properties of this new class and present many special cases. In particular, we provide an explicit formula for the distribution of the implied partition, as well as the posterior characterization of the new process in terms of the superposition of two discrete measures. We also provide consistency results. Moreover, we design both a marginal and a conditional algorithm for finite mixture models with a random number of components. These schemes are based on an auxiliary variable MCMC, which allows handling the otherwise intractable posterior distribution and overcomes the challenges associated with the Reversible Jump algorithm. We illustrate the performance and the potential of our model in a simulation study and on real data applications.