Tree-based reparameterization framework for analysis of sum-product and related algorithms
We present a tree-based reparameterization (TRP) framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation (BP) or sum-product algorithm as well as variations and extensions of BP. Algorithms in this class can be formulated as a sequence of reparameterization updates, each of which entails refactorizing a portion of the distribution corresponding to an acyclic subgraph (i.e., a tree, or more generally, a hypertree). The ultimate goal is to obtain an alternative but equivalent factorization using functions that represent (exact or approximate) marginal distributions on cliques of the graph. Our framework highlights an important property of the sum-product algorithm and the larger class of reparameterization algorithms: the original distribution on the graph with cycles is not changed. The perspective of tree-based updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency. We develop interpretations of these results in terms of information geometry. The invariance of the distribution, in conjunction with the fixed-point characterization, enables us to derive an exact expression for the difference between the true marginals on an arbitrary graph with cycles, and the approximations provided by belief propagation. More broadly, our analysis applies to any algorithm that minimizes the Bethe free energy. We also develop bounds on the approximation error, which illuminate the conditions that govern their accuracy. Finally, we show how the reparameterization perspective extends naturally to generalizations of BP (e.g., Kikuchi (1951) approximations and variants) via the notion of hypertree reparameterization.