| In this dissertation, we consider probabilistic inference problems on binary pairwise Gibbs random fields (BPW-GRFs), which belong to a class of Markov random fields with applications to a large variety of systems, including computer vision, statistical mechanics, modeling of neural functions, and others. In particular, we study the application of iterative heuristic sum-product algorithms (SPAs) to the underlying graphs for solving the marginal problem on BPW-GRFs. These algorithms operate on the BPW-GRF graph by propagating messages along the edges and by using them to update the beliefs at each node of the graph; these beliefs then serve as suboptimal solutions to the marginal problem. SPAs offer several advantages such as complexity that is polynomial in the number of nodes and edges in the graph and the ability to operate in a distributed fashion (determined by the structure of the underlying graph).;In general, the analysis of SPAs can be categorized into (i) finding conditions under which the SPAs converge, and (ii) determining the correctness of the marginal solutions provided by the SPAs with respect to the true marginals. In this dissertation, we consider both problems. For each problem, we first review existing results and then present our specific contribution within the class of BPW-GRFs. Finally, we extend our analysis of SPAs on BPW-GRFs to the application of multiple fault diagnosis (note that the equivalent GRFs for fault diagnosis systems are typically non-binary). In particular, we establish tighter bounds over previous results, and show that fault diagnosis using SPA beliefs (as suboptimal solutions to the true marginals) can detect multiple faults with very high accuracy. |
