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A comprehensive picture of three Bethe-Kikuchi variational principlesincluding their relationship to belief propagation (BP) algorithms onhypergraphs is given. The structure of BP equations is generalized to definecontinuous-time diffusions, solving localized versions of the max-entropyprinciple (A), the variational free energy principle (B), and a less usualequilibrium free energy principle (C), Legendre dual to A. Both critical pointsof Bethe-Kikuchi functionals and stationary beliefs are shown to lie at thenon-linear intersection of two constraint surfaces, enforcing energyconservation and marginal consistency respectively. The hypersurface ofsingular beliefs, accross which equilibria become unstable as the constraintsurfaces meet tangentially, is described by polynomial equations in the convexpolytope of consistent beliefs. This polynomial is expressed by a loop seriesexpansion for graphs of binary variables.

Local Max-Entropy and Free Energy Principles, Belief Diffusions and their Singularities