Bayesian Nonparametric Methods for Complex Datasets

Abstract

Modern data tend to present complex structures that challenge classical modeling assumptions and frameworks, including heterogeneity and spatial and/or temporal dependency. Bayesian nonparametric (BNP) models are a powerful tools that can address these challenges. They enable flexible modeling structures that adapt to the data complexity and provides uncertainty estimation. This dissertation proposes several BNP methods that are applicable to a wide range of statistical learning problems in regression, clustering and density estimation, with applications in fields including global health and financial econometrics. In Chapter 2, we proposed a novel model that integrates the Bayesian additive regression tree prior (BART) into the Gaussian process spatial model, aimed at spatial prediction problems where the covariate effects may be nonlinear and flexible. In Chapter 3, we studied and compared the computational performance for multivariate Hawkes processes (MHP) models, a temporal processes commonly used to model mutually exciting behaviors in temporal event sequences. In Chapter 4, we apply the dependent Dirichlet process (DDP) to model the temporal dynamics in the MHP models. Our model allows for flexible and adaptive modeling for excitation functions while borrowing information across dimensions. Future research directions related to the topic of this dissertation is outlined in Chapter 5.