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

• URL: http://arxiv.org/abs/2207.00841v1
• Date: Sat, 2 Jul 2022 14:20:40 GMT
• Title: Local Max-Entropy and Free Energy Principles Solved by Belief Propagation
• Authors: Olivier Peltre
• Abstract summary: A statistical system is classically defined on a set of microstates $E$ by a global energy function $H : E to mathbbR$, yielding Gibbs probability measures $rhobeta(H)$ for every inverse temperature $beta = T-1$. We show that the generalized belief propagation algorithm solves a collection of local variational principles, by converging to critical points of Bethe-Kikuchi approximations of the free energy $F(beta)$, the Shannon entropy $S(cal U)$, and the variational free energy
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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) $\rho^\beta(H)$ for every inverse temperature $\beta = T^{-1}$. Gibbs states are simultaneously characterized by free energy principles and the max-entropy principle, with dual constraints on inverse temperature $\beta$ and mean energy ${\cal U}(\beta) = \mathbb{E}_{\rho^\beta}[H]$ 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 $H(x) = \sum_{\rm a \subset \Omega} h_{\rm a}(x_{|\rm a})$ of local short-range interactions $h_{\rm a} : E_{\rm a} \to \mathbb{R}$ indexed by bounded subregions ${\rm a} \subset \Omega$, 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 $F(\beta)$, the Shannon entropy $S(\cal U)$, and the variational free energy ${\cal F}(\beta) = {\cal U} - \beta^{-1} S(\cal U)$, 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 $\beta$.

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