Understanding belief propagation and its generalizations

"Inference" problems arise in statistical physics, computer vision, error-correcting coding theory, and AI. We explain the principles behind the belief propagation (BP) algorithm, which is an efficient way to solve inference problems based on passing local messages. We develop a unified approach, with examples, notation, and graphical models borrowed from the relevant disciplines.We explain the close connection between the BP algorithm and the Bethe approximation of statistical physics. In particular, we show that BP can only converge to a fixed point that is also a stationary point of the Bethe approximation to the free energy. This result helps explaining the successes of the BP algorithm and enables connections to be made with variational approaches to approximate inference.The connection of BP with the Bethe approximation also suggests a way to construct new message-passing algorithms based on improvements to Bethe's approximation introduced Kikuchi and others. The new generalized belief propagation (GBP) algorithms are significantly more accurate than ordinary BP for some problems. We illustrate how to construct GBP algorithms with a detailed example.