Advances in Bayesian Model Choice

Duke University, Durham NC

Investigators

Abstract

The investigator considers new default prior distributions for Bayesian model selection and model averaging in linear and generalized linear models. Prior choice for model specific parameters and for prior model probabilities is of critical importance, particularly for modeling high dimensional relationships where subjective specifications are impractical or where an ``objective'' analysis is desired. New families of objective priors that have desirable risk properties, adapt to unknown degrees of sparsity, and also permit tractable computations for large scale model search are studied. In high dimensional problems, the dimension of the model space is astronomical. Innovative methodology and algorithms for large scale stochastic search and model averaging for high dimensional model spaces are developed, with an emphasis on algorithms that exploit the architecture of Graphical Processing Units (GPUs). GPUs provide the computing power of a distributed cluster, but at a fraction of their cost and space/cooling requirements, but require care in the development of statistical algorithms that take advantage of their architecture. Advances in technology have led to the collection of high dimensional data structures, spawning an increasingly complex array of statistical models for data. Goals of data analysis may include prediction and/or selection of a subset of models to test particular theories or to reduce attention from many speculative models to a few well chosen models; these are fundamental problems in statistics and throughout the sciences. In such settings, model uncertainty is ubiquitous. Bayesian methods offer an effective and conceptually appealing approach for addressing model uncertainty through Bayesian model averaging, incorporating both parameter and model uncertainty, while still permitting selection of a model via coherent decision-theoretic principles. The methodological developments are driven by issues that arise from the following applications 1) identifying important factors to predict protein activity; 2) predicting risk of international conflict; 3) health effects of criteria pollutants; and 4) identifying single nucleotide polymorphisms that are associated with ovarian cancer risk. As prediction and model selection are some of the most fundamental and widespread problems in the sciences and beyond, the project's impact extends beyond the applications listed above. The development and distribution of software ensures that these methods are widely available.