Low-Resolution Statistical Modeling With Belief Functions

Abstract

This dissertation addresses problems in the theoretical, methodological, and computational aspects of statistical modeling in the presence of low-resolution information. I consider a structural kind of model uncertainty that cannot be encapsulated by a single probability distribution but by a collection of them, and in a special case, by a belief function. I investigate issues that arise with the choice of conditioning rules when employing belief functions and Choquet capacities for uncertain inference, a challenge that does not exist within the realm of precise probability modeling. Under Dempster-Shafer theory, a system of Extended Probability Calculus, I showcase a novel belief function model for multinomial data as a partially identified Poisson experiment, and offer computationally efficient posterior inference in the form of stochastic three-valued logic. Numerical and sequential Monte Carlo techniques are discussed within the contexts of vaccine efficacy and HIV-1 infection timing studies.