| In this thesis we examine the problem of uniqueness of Gibbs measures. To this end, we develop new techniques for bounding distances between probability measures using partial information about the conditional distributions. The results are applied to a set of interrelated questions regarding convergence of Gibbs samplers and the Sum-Product algorithm.;The common theme linking all the applications is the construction of joint distributions, called Gibbs measures, for finite or countably infinite collections of discrete random variables. Gibbs samplers are Markov chains used to generate observations from finite Gibbs measures; the Sum-Product algorithm iteratively computes marginals of finite Gibbs measures. We present conditions for rapid convergence of both algorithms. For Sum-Product, this investigation leads to the study of computation trees and infinite Gibbs measures.;Chapter 2 is a survey of the many methods developed by Dobrushin and later authors to establish rapid mixing of Gibbs samplers and uniqueness of infinite Gibbs measures. Here our contribution is one of unification. Specifically, we present a systematic reduction of a large collection of conditions to a set of contraction properties for various operators.;In Chapter 3, our main contribution is a generic technique to strengthen existing conditions for uniqueness of the infinite Gibbs measure. This leads to a new uniqueness criterion, which we call multi-hop Dobrushin condition.;In Chapter 4, we apply the multi-hop Dobrushin condition to obtain new convergence results for the Sum-Product algorithm on loopy factor graphs. In particular, we describe algorithms that check the multi-hop Dobrushin condition on computation trees. |
