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Asymptotic analysis and dependent construction of Bayesian nonparameric models
- Author / Creator
- Zhang, Junxi
Bayesian nonparametric models have gained increasing attention due to their flexibility in modelling natural and social phenomena and have been widely applied in machine learning, biology, social science and so on. Unlike traditional Bayesian parametric models, Bayesian nonparametric models place priors on an infinite dimensional space and allow the model itself to be determined by data. To understand and apply Bayesian nonparametric models, the properties, especially the asymptotic analysis, of the priors and posteriors of Bayesian nonparametric models should be studied. In this dissertation, various asymptotic problems for Bayesian nonparametric priors and posteriors are studied, and two dependent Bayesian nonparametric priors are constructed. This thesis includes three main parts corresponding to three papers. In the first part, we obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general iid stick-breaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized
Dirichlet process. For the stick-breaking process with general iid stick-breaking weights, two general conditions are formulated such that the asymptotic theorems hold. In the
second part, we present the posterior consistency analysis for normalized random measures with independent increments (NRMIs) through the corresponding Levy intensities, which can be used to characterize the completely random measures in the construction of NRMIs. An assumption based on the Levy intensities for analysing the posterior consistency of NRMIs is introduced and verified with multiple examples. Furthermore, we derive the Bernstein-von Mises theorem for the normalized generalized gamma process, based on which, credible intervals are constructed with some discussions and numerical illustration. In the third part, we construct two classes of dependent Bayesian nonparametric models through the normalization of completely random measures driven by Cox processes. We provide multiple distribution theories for the two constructions including moments, probabilistic characterizations of the induced random partition structures by the hierarchical models, distributions of the random partition numbers.
- Graduation date
- Fall 2023
- Type of Item
- Doctor of Philosophy
- This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.