Contributions to dependent processes in Bayesian nonparametrics

Random measures represent the fundamental building blocks for defining flexible priors in Bayesian nonparametric models. Over the last three decades, there has been a widespread diffusion of proposals aimed at introducing dependence among different random measures, in order to properly account for various forms of heterogeneity in the observations while preserving the possibility to borrow information across them. The first part of the thesis focuses on vectors of dependent random measures defined through hierarchical structures, arguably the most natural and popular strategy to specify nonparametric priors for partially exchangeable data. These prior processes induce a random nested partition structure on the observations, which is usually taken into account by introducing additional latent variables. In this work, we propose a unified approach to hierarchies of random measures, in which such latent variables are directly inserted into the generative model for the data; within this framework, we identify a common structure shared by different hierarchical constructions proposed in the literature, and highlight a key identity which plays a prominent role in the derivation of quantities of interest. Furthermore, we consider hierarchical completely random measures as mixing measures to define dependent mixture hazard rates, which are in turn employed in the context of survival analysis to model competing risks data. We derive analytical results for Bayesian inference and prediction, as well as efficient posterior sampling algorithms; the effectiveness of our proposal is tested on simulated and clinical datasets. The second part of the thesis contributes to the Bayesian nonparametric regression framework, by introducing a collection of dependent random probability measures indexed by covariates, which enter the model specification through a multiplicative kernel structure. This construction induces a predictor-dependent random partition model, characterized by great flexibility and inherent consistency for new observations, while retaining some analytical tractability. A noteworthy example arises when the distribution of such random measures is a transformation of the distribution of a stable process; moreover, the structure of the posterior distribution implied by such specification suggests the introduction of a novel nonhomogeneous process, which extends the two-parameter Poisson-Dirichlet process and acts as quasi-conjugate prior for our proposal. In the last chapter, we further develop this novel nonparametric process in the exchangeable setting, and characterize its prior-posterior updating mechanism, as well as its predictive structure.