  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 $\beta$.
