Tree-based reparameterization analysis of belief propagation and related algorithms for approximate inference on graphs with cycles

Although it is straightforward to compute the marginals of a distribution p(x) defined by a tree-structured graphical model, this same task is often difficult for graphs with cycles. The belief propagation (BP) or sum-product algorithm is an approximate method for computing such marginals; it is used in various applications (e.g., iterative decoding of turbo and LDPC codes). Belief propagation is typically presented and analyzed as a sequence of message-passing operations. We develop a different conceptual perspective that involves reparameterizing the original distribution in terms of so-called pseudomarginals on cliques of the graph. This view gives rise to a simple characterization of the fixed points, as well as an exact expression and bounds on the error for an arbitrary graph with cycles.