Chegini, Sadegh2025-04-242025-04-242025-04-24http://hdl.handle.net/10393/50372https://doi.org/10.20381/ruor-31044Bayesian nonparametric inference requires the construction of priors on infinite-dimensional spaces, such as the space of cumulative distribution functions. Well-known priors on this space include the Dirichlet process and the two-parameter Poisson–Dirichlet process. In this thesis, we explore a distinctive functional of the Poisson point process, known as the negative binomial process. While the increments of the negative binomial process are not independent, they become conditionally independent given an underlying gamma variable. We propose a novel point process representation for the negative binomial process, which extends the Poisson-Kingman distribution and its associated random discrete probability measure. The new proposed family of the discrete random probability measures which is defined by normalizing the points of the negative binomial process provides a new set of useful priors for Bayesian nonparametric models with more flexibility compared to the random discrete probability measure which are obtained by normalizing the points of a Poisson point process. We illustrate how this family encompasses several well-known priors, such as the Dirichlet process, the normalized positive α-stable process, and the Poisson–Dirichlet process. Using the same gamma Lévy measure, we derive an extension of the Dirichlet process along with an almost sure approximation. Additionally, leveraging our negative binomial process representation, we develop a new series representation for the Poisson–Dirichlet process. Through simulations, we demonstrate how adopting priors from this family can enhance the performance of Bayesian nonparametric hierarchical models. In the literature, the term negative binomial process has been used to describe several distinct stochastic processes, each playing a significant role in probability theory and statistics, particularly in Bayesian nonparametric analysis. However, the presence of multiple, and at times conflicting, definitions has led to considerable ambiguity. This thesis addresses this issue by systematically reviewing the various definitions and clarifying their distinctions. The aim is to provide a comprehensive overview that helps practitioners recognize the differences between these processes and avoid potential misunderstandings. Furthermore, for one of the definitions of the negative binomial process, we present an extension from the univariate case to a bivariate form. We also examine the Liouville distribution, a well-known conjugate prior for the multinomial distribution, which addresses certain limitations of the Dirichlet distribution, particularly its tendency to induce negative correlations. We construct a discrete random probability measure based on a random vector following a Liouville distribution and establish its weak limit to define the proposed Liouville process. This process takes the form of a spike-and-slab model, where the slab is represented by a Dirichlet process and the spike corresponds to a single point drawn from its mean. These components are combined through a random convex mixture, with weights governed by the Liouville distribution. By placing the Liouville process as a prior over the space of probability measures, we derive both its posterior and predictive distributions.enBayesian nonparametricα-stable processnegative binomial processLiouville processDirichlet processPoisson-Kingman distributiontwo-parameter Poisson–Dirichlet processThesis