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.

给出了对三个Bethe-Kikuchi变分原理的全面描述，包括它们与信念传播算法在超图上的关系。将信念传播方程的结构推广到定义连续时间扩散，解决最大熵原理(A)、变分自由能原理(B)和一个不太常见的Legendre对偶于A的平衡自由能原理(C)中的局部版本。Bethe-Kikuchi泛函的临界点和稳态信念都被证明位于两个约束曲面的非线性交点上，分别强制能量守恒和边缘一致性。描述了奇异信念的超曲面，当约束曲面以切线相遇时，平衡变得不稳定，这个超曲面由一阶二进制变量图的多项式方程在一致信念的凸多胞体中表达。

局部最大熵与自由能原理，信念扩散及其奇异性