| Tóm tắt: | <p>The Bayesian methodology provides a compelling framework that offers an intuitive yet practical approach to model the world, allowing refinement of the statistician’s prior knowledge in light of incoming empirical data. Any unknown quantity is regarded as random, and its updated knowledge, called the posterior distribution, quantifies the remaining uncertainty therein. The choice of the prior distribution is at the core of Bayesian modeling and plays a primary role in the model’s performance. Bayesian Nonparametric (BNP) modeling utilizes infinitely many parameters to capture intricate patterns of real-world phenomena and adapts to the complexity of the data.</p>
<p>Bayesian Nonparametrics is, therefore, an appealing tool that has found many successful applications in a large variety of domains such as machine learning, ecology, biology, finance, etc. Our research focus is on modeling and analyzing discrete structures, with a particular interest in the context of clustering and feature allocation. In the first introductory chapter we review the necessary theoretical background and present the critical building blocks for the constructions proposed in the following chapters. The fist part of this dissertation is dedicated to modeling complex data with Bayesian Nonparametrics. Our objective is to overcome the limitations of existing approaches and describe models that can capture real-world characteristics. We pay careful attention to power-law properties, which seem to be ubiquitous in real-world applications, and propose models for networks with community structures, clustering, and financial time-series. In the second part, we move beyond modelling purposes and show that Bayesian nonparametric can offer powerful tools for the theoretical analysis of (non-Bayesian) estimation problems. The part is dedicated to the question of rare events and missing mass estimation in the species sampling and feature allocation frameworks. Using carefully chosen Nonparametric priors, we show the nonexistence of universally consistent estimators of the missing mass. We also derive lower bounds on rates of estimation of rare events under appropriate assumptions.</p>
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