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Local Max-Entropy and Free Energy Principles Solved by Belief Propagation

A statistical system is classically defined on a set of microstates $E$ by a global energy function $H : E \to \mathbb{R}$, yielding Gibbs probability measures (softmins) $ρ^β(H)$ for every inverse temperature $β= T^{-1}$. Gibbs states are simultaneously characterized by free energy principles and the max-entropy principle, with dual constraints on inverse temperature $β$ and mean energy ${\cal U}(β) = \mathbb{E}_{ρ^β}[H]$ respectively. The Legendre transform relates these diverse variational principles which are unfortunately not tractable in high dimension. The global energy is generally given as a sum $H(x) = \sum_{\rm a \subset Ω} h_{\rm a}(x_{|\rm a})$ of local short-range interactions $h_{\rm a} : E_{\rm a} \to \mathbb{R}$ indexed by bounded subregions ${\rm a} \subset Ω$, and this local structure can be used to design good approximation schemes on thermodynamic functionals. We show that the generalized belief propagation (GBP) algorithm solves a collection of local variational principles, by converging to critical points of Bethe-Kikuchi approximations of the free energy $F(β)$, the Shannon entropy $S(\cal U)$, and the variational free energy ${\cal F}(β) = {\cal U} - β^{-1} S(\cal U)$, extending an initial correspondence by Yedidia et al. This local form of Legendre duality yields a possible degenerate relationship between mean energy ${\cal U}$ and $β$.