Modelling with discrete random probability measures

Random probability measures are a cornerstone of Bayesian nonparametrics. By virtue of de Finetti's representation theorem, their law acts as the prior distribution for exchangeable observations. Mostly used Bayesian nonparametric procedures, in this framework, rely on laws selecting almost surely discrete probability measures, such as the celebrated Dirichlet process and its several extensions. The first part of this thesis is dedicated to problems related to random probability measures arising in the exchangeable regime. We explore properties of functionals of noteworthy discrete random probability measures in order to provide prior elicitation. In particular we retrieve explicit expressions for base measures inducing a broad class of distributions on the random mean of a Dirichlet process, a normalized stable process and a Pitman--Yor process. We furthermore provide an application to widely employed mixture models. These results have led us to further theoretical investigations regarding the connection between Dirichlet random means and continual Young diagrams. The second part of the thesis is instead devoted to the partially exchangeable regime, a generalization of exchangeability which encompasses a more complex dependence structure among observations naturally divided in groups. We rely on hierarchical discrete random probability measures to enforce such distributional invariance in a model for clustering of nodes in multilayer networks. The induced distribution on the space of sequences of consistent partitions, determined by partially exchangeable partition probability functions, allows for theoretically validated prediction regarding new nodes incoming into the network.