Confronto di modelli di mistura infinita location e location-scale per stima bayesiana di densità

Dirichlet process mixture models are Bayesian nonparametric models that are widely used in machine learning as clustering models and in Statistics for density estimation. Two of these models will be described and compared: location and location-scale infinite mixture models. Given a set of observations distributed according an unknown distribution, these two models give an approximation of the density which consists in an infinite linear combination of Gaussian densities where both parameters and coefficients are random variables. As the models' names suggest, the difference between them is about the scale parameter. In the location-scale model both location vectors and covariance matrices are in the mixture, while in the location model the mean is involved but the scale parameter is shared by all the mixture components and it is treated as a hyperparameter. It is intuitive that the location-scale model gives a more accurate estimation since it is free to assign different covariance matrices to its components. Moreover, in the cluster problem the location-scale model must be more parsimonious in estimating the number of clusters. For these reasons, many practitioners have used this model in multivariate density estimation although asymptotic results had been proved just for the multivariate location mixtures. The first chapter introduces the basic notions and results of the Bayesian statistics and Bayesian nonparametrics, in particular we will define the Dirichlet process which is essential to construct the two models above mentioned. In the second chapter, firstly we will explore the theory behind the algorithms that have been implemented to compute the density estimations, i.e. the theory of the ergodic Markov chains; secondly there will be a review of the most famous Gibbs sampling algorithms used in Bayesian statistics to sample from posterior distributions, in particular the algorithm ``Neal 8" which will be adopted for the simulations done in the last chapter.