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In this paper we treat both forms of probabilistic inference, estimatingmarginal probabilities of the joint distribution and finding the most probableassignment, through a unified message-passing algorithm architecture. Wegeneralize the Belief Propagation (BP) algorithms of sum-product andmax-product and tree-rewaighted (TRW) sum and max product algorithms (TRBP) andintroduce a new set of convergent algorithms based on "convex-free-energy" andLinear-Programming (LP) relaxation as a zero-temprature of aconvex-free-energy. The main idea of this work arises from taking a generalperspective on the existing BP and TRBP algorithms while observing that theyall are reductions from the basic optimization formula of $f + \sum_i h_i$where the function $f$ is an extended-valued, strictly convex but non-smoothand the functions $h_i$ are extended-valued functions (not necessarily convex).We use tools from convex duality to present the "primal-dual ascent" algorithmwhich is an extension of the Bregman successive projection scheme and isdesigned to handle optimization of the general type $f + \sum_i h_i$. Mappingthe fractional-free-energy variational principle to this framework introducesthe "norm-product" message-passing. Special cases include sum-product andmax-product (BP algorithms) and the TRBP algorithms. When thefractional-free-energy is set to be convex (convex-free-energy) thenorm-product is globally convergent for estimating of marginal probabilitiesand for approximating the LP-relaxation. We also introduce another branch ofthe norm-product, the "convex-max-product". The convex-max-product isconvergent (unlike max-product) and aims at solving the LP-relaxation.

Norm-Product Belief Propagation: Primal-Dual Message-Passing for Approximate Inference