la rappresentazione stick-breaking di un processo Poisson-Kingman 1/2 stable

Stick-breaking representation for ½-stable Poisson-Kingman process priors Bayesian nonparametrics has recently undergone a strong development in terms of both theory and practice. At the heart of the approach there is the concept of random probability measure whose distribution acts as a prior distribution for Bayesian nonparametric inference, the most notable example being the Dirichlet process prior introduced by Ferguson (1973). Many well-known priors used in practice admit different, though equivalent in distribution, representations. Some of these are convenient if one wishes to thoroughly analyze the theoretical properties of the priors being used, others are more useful in terms of modelling and computation. In terms of the latter, the socalled stick-breaking constructions certainly stand out. Indeed, they allow to define efficient simulation algorithms and, furthermore, they represent a useful tool for the construction of nonparametric priors capable of incorporating certain forms of dependence for the observables and hence, go beyond the standard exchangeability setting. In my thesis I focus on the class of ½-stable Poisson-Kingman process priors introduced by Pitman (2003), which includes as special cases several priors currently used in Bayesian nonparametrics. Such a class of priors is defined hierarchically in terms of an underlying normalized ½-stable completely random measure which is suitably mixed, with respect to some distribution G, over its normalizing total mass. In other terms, a ½-stable Poisson-Kingman process prior is a random probability measure parameterized by a distribution G over the positive real line. By exploiting such a hierarchical representation, I introduce a novel constructive definition for the class of ½-stable Poisson-Kingman process prior by providing a completely explicit stick-breaking representation of the conditional underlying random probability measure given the normalizing total mass with distribution G. The proposed constructive definition allows to recover as special cases the stickbreaking representation of some well-known nonparametric priors such as, the two parameter Poisson-Dirichlet process prior with discount parameter ½ and the normalized inverse Gaussian process prior. But new priors could be obtained by a particular choice of the mixing distribution G for the total mass. The proposed stick-breaking representation is interesting from both theoretical, modelling and computational points of view. Firstly, it completes the study of the well-known class of ½-stable Poisson-Kingman process priors, by providing a novel constructive definition in terms of a latent random variable with distribution G and a collection of dependent stick-breaking weights defined by means of a suitable transformation involving Gamma random variables and inverse Gamma random variables. Secondly, it suggests a simple way to define new Bayesian nonparametric models based on the class of ½-stable Poisson-Kingman process priors by modifying well-established models based on the stick-breaking representation of the Dirichlet process prior. Finally, the proposed stick-breaking representation allows to extend to the class of ½-stable Poisson-Kingman process priors various recently proposed simulation schemes, such as the blocked Gibbs sampling algorithm, the slice sampling algorithm and the retrospective sampling algorithm. Indeed, these simulation algorithms assumes to have access to a stick breaking representati