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 → ℝ, yielding Gibbs probability measures (softmins) ρβ(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) = mathbbEρβ[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) = sumrm a subset Omega hrm a(x|rm a) of local short-range interactions hrm a : Erm a → 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(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.