Modelli mistura in statistica Bayesiana nonparametrica: l'algoritmo Ordered Allocation Sampler

Nonparametric Bayesian methods have become quite popular in the last twenty years, thanks to all the successful applications they found in the literature. In particular, one of the most promising and powerful techniques of Bayesian Statistics are mixture models, which allow to formalize through a probabilistic framework the presence of sub-groups inside a population of interest. They have largely been employed for clustering, classification and density estimation. The aim of this work is to provide an introduction to mixture models, explaining the differences between frequentist and Bayesian estimation and finally we focus on the non parametric Bayesian modeling. In particular we discuss species sampling models as priors for the mixing distributions. After a theoretical presentation, a few algorithms are described in detail, with a particular focus on the “Ordered Allocation Sampler”, an original Gibbs sampler proposed in a paper by De Blasi and Gil-Leyva, followed by the comparison of their performances and some standard examples. Finally, a real application is provided, in order to explore the true potential of these techniques, showing how they can be applied to some modern problems, like the prediction of the arising of Alzheimer disease, with a survival analysis enhanced by an initial clustering of the patients.