On Dependent Processes in Bayesian Nonparametrics: Theory, Methods, and Applications

The main topics of the thesis are dependent processes and their uses in Bayesian nonparametric statistics. With the term dependent processes, we refer to two or more infinite dimensional random objects, i.e., random probability measures, completely random measures, and random partitions, whose joint probability law does not factorize and, thus, encodes non-trivial dependence. We investigate properties and limits of existing nonparametric dependent priors and propose new dependent processes that fill gaps in the existing literature. To do so, we first define a class of priors, namely multivariate species sampling processes, which encompasses many dependent processes used in Bayesian nonparametrics. We derive a series of theoretical results for the priors within this class, keeping as main focus the dependence induced between observations as well as between random probability measures. Then, in light of our theoretical findings, as well as considering specific motivating applications, we develop novel prior processes outside this class, enlarging the types of data structures and prior information that can be handled by the Bayesian nonparametric approach. We propose three new classes of dependent processes: full-range borrowing of information priors, invariant dependent priors (with a focus on symmetric hierarchical Dirichlet processes), and dependent priors for panel count data. Full-range borrowing of information priors are dependent random probability measures that may induce either positive or negative correlation across observations and, thus, they achieve high flexibility in the type of induced dependence. Moreover, they introduce an innovative idea of borrowing of information across samples which differs from classical shrinkage. Invariant dependent priors are instead dependent random probabilities that almost surely satisfy a specified invariance condition, e.g., symmetry. They may be employed both when a priori knowledge on the shape of the unknown distribution is available or, as we do, to flexibly model errors terms in complex models without losing identifiability of other parameters of interest. Finally, dependent priors for panel count data are flexible priors based on completely random measures, that take into account dependence between the observed counts and the frequency of observation in panel count data studies. We study a priori and a posteriori properties of all the proposed models, develop algorithms to derive inference, compare the performances of our proposals with existing methods, and apply these constructions to simulated and real datasets. Through all the thesis, we try to balance theoretical and methodological results with real-world applications.