Finite-dimensional nonparametric priors: theory and applications

The investigation of flexible classes of discrete prior has been an active research line in Bayesian statistics. Several contributions were devoted to the study of nonparametric priors, including the Dirichlet process, the Pitman–Yor process and normalized random measures with independent increments (NRMI). In contrast, only few finite-dimensional discrete priors are known, and even less come with sufficient theoretical guarantees. In this thesis we aim at filling this gap by presenting several novel general classes of parametric priors closely connected to well-known infinite-dimensional processes, which are recovered as limiting case. A priori and posteriori properties are extensively studied. For instance, we determine explicit expressions for the induced random partition, the associated urn schemes and the posterior distributions. Furthermore, we exploit finite-dimensional approximations to facilitate posterior computations in complex models beyond the exchangeability framework. Our theoretical and computational findings are employed in a variety of real statistical problems, covering toxicological, sociological, and marketing applications.