论文标题
通过信仰传播解决的局部最大 - 凝集和自由能原则
Local Max-Entropy and Free Energy Principles Solved by Belief Propagation
论文作者
论文摘要
全球能量函数$ h：e \ to \ mathbb {r} $在一组微晶体上定义了一个统计系统，从而产生gibbs概率度量（softmins）$ρ^β（h）$ $ρ^β（h）$对于每个相反温度$β= T^{ - 1} $。 Gibbs状态同时以自由能原理和最大原理为特征，在反温度$β$上具有双重约束，平均能量$ {\ cal U}（β）= \ Mathbb {e} _ {ρ^β} [H] $。 Legendre Transform关联了这些多样化的变异原则，不幸的是，在高维度中无法处理。 全局能量通常以$ h（x）= \ sum _ {\ rm a \ subsetω} h _ {\ rm a}（x_ {| \ rm a}）$ $ {\ rm a} \子集ω$，并且该局部结构可用于设计热力学功能上的良好近似方案。 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 ${\cal F}(β) = {\cal U} - β^{-1} s（\ cal u）$，扩展了Yedidia等人的初始通信。这种局部形式的legendre二重性产生了平均能量$ {\ cal u} $和$β$之间可能的退化关系。
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 $β$.
