Generalized Empirical Bayes: Theory, Methodology, and Applications

Abstract

The two key issues of modern Bayesian statistics are: (i) establishing a principled approach for \textit{distilling} a statistical prior distribution that is \textit{consistent} with the given data from an initial believable scientific prior; and (ii) development of a \textit{consolidated} Bayes-frequentist data analysis workflow that is more effective than either of the two separately. In this thesis, we propose generalized empirical Bayes as a new framework for exploring these fundamental questions along with a wide range of applications spanning fields as diverse as clinical trials, metrology, insurance, medicine, and ecology. Our research marks a significant step towards bridging the ``gap'' between Bayesian and frequentist schools of thought that has plagued statisticians for over 250 years. Chapters 1 and 2---based on \cite{mukhopadhyay2018generalized}---introduces the core theory and methods of our proposed generalized empirical Bayes (gEB) framework that solves a long-standing puzzle of modern Bayes, originally posed by Herbert Robbins (1980). One of the main contributions of this research is to introduce and study a new class of nonparametric priors ${\rm DS}(G, m)$ that allows exploratory Bayesian modeling. However, at a practical level, major practical advantages of our proposal are: (i) computational ease (it does not require Markov chain Monte Carlo (MCMC), variational methods, or any other sophisticated computational techniques); (ii) simplicity and interpretability of the underlying theoretical framework which is general enough to include almost all commonly encountered models; and (iii) easy integration with mainframe Bayesian analysis that makes it readily applicable to a wide range of problems. Connections with other Bayesian cultures are also presented in the chapter. Chapter 3 deals with the topic of measurement uncertainty from a new angle by introducing the foundation of nonparametric meta-analysis. We have applied the proposed methodology to real data examples from astronomy, physics, and medical disciplines. Chapter 4 discusses some further extensions and application of our theory to distributed big data modeling and the missing species problem. The dissertation concludes by highlighting two important areas of future work: a full Bayesian implementation workflow and potential applications in cybersecurity.