2023/10/04 by Olivier Peltre, Peltre, Olivier
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #05-02 #55N25 #55U10 #94A17 #Algebraic Topology (math.AT) #Artificial Intelligence (cs.AI) #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #G.2.1 #G.2.2 #Gene Regulatory Network Analysis #I.2.0 #Information Theory (cs.IT) #Mathematical Biology Tumor Growth #Mathematical Physics (math-ph) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2310.02946
openalex publication_date 2023/10/04 · openalex created_date 2023/10/06 · openalex updated_date 2026/08/01
A comprehensive picture of three Bethe-Kikuchi variational principles including their relationship to belief propagation (BP) algorithms on hypergraphs is given. The structure of BP equations is generalized to define continuous-time diffusions, solving localized versions of the max-entropy principle (A), the variational free energy principle (B), and a less usual equilibrium free energy principle (C), Legendre dual to A. Both critical points of Bethe-Kikuchi functionals and stationary beliefs are shown to lie at the non-linear intersection of two constraint surfaces, enforcing energy conservation and marginal consistency respectively. The hypersurface of singular beliefs, accross which equilibria become unstable as the constraint surfaces meet tangentially, is described by polynomial equations in the convex polytope of consistent beliefs. This polynomial is expressed by a loop series expansion for graphs of binary variables.