On Loopy Belief Propagation - Local Stability Analysis for Non-Vanishing Fields

In this work we obtain all fixed points of belief propagation and perform a local stability analysis. We consider pairwise interactions of binary random variables and investigate the influence of non-vanishing fields and finite-size graphs on the performance of belief propagation; local stability is heavily influenced by these properties. We show why non-vanishing fields help to achieve convergence and increase the accuracy of belief propagation. We further explain the close connections between the underlying graph structure, the existence of multiple solutions, and the capability of belief propagation (with damping) to converge. Finally we provide insights into why finite-size graphs behave better than infinite-size graphs.