Advances in Empirical Bayes Modeling and Bayesian Computation

Abstract

Chapter 1 of this thesis focuses on accelerating perfect sampling algorithms for a Bayesian hierarchical model. A discrete data augmentation scheme together with two different parameterizations yields two Gibbs samplers for sampling from the posterior distribution of the hyperparameters of the Dirichlet-multinomial hierarchical model under a default prior distribution. The finite-state space nature of this data augmentation permits us to construct two perfect samplers using bounding chains that take advantage of monotonicity and anti-monotonicity in the target posterior distribution, but both are impractically slow. We demonstrate however that a composite algorithm that strategically alternates between the two samplers' updates can be substantially faster than either individually. We theoretically bound the expected time until coalescence for the composite algorithm, and show via simulation that the theoretical bounds can be close to actual performance. Chapters 2 and 3 introduce a strategy for constructing scientifically sensible priors in complex models. We call these priors catalytic priors to suggest that adding such prior information catalyzes our ability to use richer, more realistic models. Because they depend on observed data, catalytic priors are a tool for empirical Bayes modeling. The overall perspective is data-driven: catalytic priors have a pseudo-data interpretation, and the building blocks are alternative plausible models for observations, yielding behavior similar to hierarchical models but with a conceptual shift away from distributional assumptions on parameters. The posterior under a catalytic prior can be viewed as an optimal approximation to a target measure, subject to a constraint on the posterior distribution's predictive implications. In Chapter 3, we apply catalytic priors to several familiar models and investigate the performance of the resulting posterior distributions. We also illustrate the application of catalytic priors in a preliminary analysis of the effectiveness of a job training program, which is complicated by the need to account for noncompliance, partially defined outcomes, and missing outcome data.