A comprehensive picture of three Bethe-Kikuchi variational principles including their relationship to belief propagation (BP) algorithms on hypergraphs is given. The structure of BP equations is generalized to define continuous-time diffusions, solving localized versions of the max-entropy principle (A), the variational free energy principle (B), and a less usual equilibrium free energy principle (C), Legendre dual to A. Both critical points of Bethe-Kikuchi functionals and stationary beliefs are shown to lie at the non-linear intersection of two constraint surfaces, enforcing energy conservation and marginal consistency respectively. The hypersurface of singular beliefs, accross which equilibria become unstable as the constraint surfaces meet tangentially, is described by polynomial equations in the convex polytope of consistent beliefs. This polynomial is expressed by a loop series expansion for graphs of binary variables.

翻译：全面阐述了三种Bethe-Kikuchi变分原理及其与超图上置信传播（BP）算法的关系。将BP方程的结构推广，定义了连续时间扩散过程，分别求解了最大熵原理（A）、变分自由能原理（B）以及A的Legendre对偶——一种较少见的平衡自由能原理（C）的局部化版本。研究表明，Bethe-Kikuchi泛函的临界点与平稳信念均位于两个约束曲面的非线性交点上，这两个曲面分别强制执行能量守恒与边缘一致性。奇异信念的超曲面——当约束曲面切向相交时，平衡态在该曲面处变得不稳定——由一致信念凸多胞体中的多项式方程描述。针对二元变量图，该多项式通过环路级数展开式表达。