Tree-based reparameterization for approximate inference on loopy graphs
We develop a tree-based reparameterization framework that provides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles. It includes belief propagation (BP), which can be reformulated as a very local form of reparameterization. More generally, we consider algorithms that perform exact computations over spanning trees of the full graph. On the practical side, we find that such tree reparameterization (TRP) algorithms have convergence properties superior to BP. The reparameterization perspective also provides a number of theoretical insights into approximate inference, including a new characterization of fixed points; and an invariance intrinsic to TRP/BP. These two properties enable us to analyze and bound the error between the TRP/BP approximations and the actual marginals. While our results arise naturally from the TRP perspective, most of them apply in an algorithm-independent manner to any local minimum of the Bethe free energy. Our results also have natural extensions to more structured approximations [e.g., 1, 2].