Paradoxes and Priors in Bayesian Regression
Agniva Som · OhioLink ETD Center (Ohio Library and Information Network) · 2014
The linear model has been by far the most popular and most attractive choice of a statistical model over the past century, ubiquitous in both frequentist and Bayesian literature.The basic model has been gradually improved over the years to deal with stronger features in the data like multicollinearity, non-linear or functional data patterns, violation of underlying model assumptions etc.One valuable direction pursued in the enrichment of the linear model is the use of Bayesian methods, which blend information from the data likelihood and suitable prior distributions placed on the unknown model parameters to carry out inference.This dissertation studies the modeling implications of many common prior distributions in linear regression, including the popular g prior and its recent ameliorations.Formalization of desirable characteristics for model comparison and parameter estimation has led to the growth of appropriate mixtures of g priors that conform to the seven standard model selection criteria laid out by Bayarri et al. (2012).The existence of some of these properties (or lack thereof) is demonstrated by examining the behavior of the prior under suitable limits on the likelihood or on the prior itself.The first part of the dissertation introduces a new form of an asymptotic limit, the conditional information asymptotic, driven by a situation arising in many practical problems when one or more groups of regression coefficients are much larger than the rest.Under this asymptotic, many prominent "g-type" priors are shown to suffer from two new unsatisfactory behaviors, the Conditional Lindley's Paradox and Essentially