On the Relationship Between the Minimum of the Bethe Free Energy Function of a Factor Graph and Sum-Product Algorithm Fixed Points

The sum-product algorithm (SPA) is a popular algorithm for efficiently approximating the marginals and the partition function of a factor graph. Some key results for this algorithm were established by Yedidia et al., who proved that, roughly speaking, fixed points of the SPA correspond to stationary points of the Bethe free energy function. However, some of their results were only for factor graphs where the local functions take on strictly positive values. They also conjectured that similar results hold for factor graphs where the local functions take on non-negative values. In this paper we make progress toward resolving this conjecture. In particular, we present examples where the results of Yedidia et al. generalize and examples where their results do not generalize. Finally, we present a general framework for analyzing fixed-points of the SPA based on a suitable dualization of the Bethe free energy function.