Book Description

In this dissertation, we first develop a novel perspective to compare Bayesian variable selection procedures in terms of their selection criteria as well as their finite-sample properties. Secondly, we investigate Bayesian post-selection inference in two types of selection problems: linear regression and population selection. We will demonstrate that both inference problems are susceptible to selection effects since the selection procedure is data-dependent. Before comparing Bayesian variable selection procedures, we first classify the current Bayesian variable selection procedures into two classes: those with selection criteria defined on the space of candidate models, and those with selection criteria not explicitly formulated on the model space. For selection methods which do not operate on the model space, it is not obvious or well-established how to assess Bayesian selection consistency. By comparing their selection criteria, we establish connections between these classes of selection methods to facilitate discussion of Bayesian variable selection consistency for both classes. Moreover, The former group can be further divided into two sub-classes depending on their use of either the Bayes Factor (BF) or estimates of marginal inclusion probabilities. In the context of linear regression, we first consider the finite sample properties of Bayesian variable selection procedures, focusing on their associated selection uncertainties and their respective empirical frequencies of correct selection, across a broad range of data generating processes. Then we consider Bayesian inference after Bayesian variable selection. Since this type of study is completely new in the Bayesian literature, we must first address many conceptual difficulties in inference after Bayesian variable selection, and more generally Bayesian inference for different types of target parameters that are relevant to the setting of Bayesian variable selection. We give some analytic arguments and simulation-based evidence to illustrate some of the possible selection effects. For population selection problem, we propose a decision-theoretical way to investigate its post-selection inference. In particular, we focus on credible intervals. When the task is to select the best population and construct a credible interval simultaneously, a compound loss function is proposed. We then derive the corresponding Bayes rule, which has both intuitive and theoretical appeal.