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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >Partition function loop series for a general graphical model: Free-energy corrections and message-passing equations
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Partition function loop series for a general graphical model: Free-energy corrections and message-passing equations

机译：通用图形模型的分区函数循环系列：自由能校正和消息传递方程

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摘要

A loop series expansion for the partition function of a general statistical model on a graph is carried out. If the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief-propagation equation, the first term of the loop series gives the BethePeierls (BP) free-energy functional at the replica-symmetric level of the mean-field spin-glass theory, and corrections are contributed only by subgraphs that are free of dangling edges. This result generalizes the early work of Chertkov and Chernyak on binary statistical models. If the belief-propagation equation has multiple fixed points, a loop series expansion is performed for the grand partition function. The first term of this series gives the BP free-energy functional at the first-step replica-symmetry-breaking (RSB) level of the mean-field spin-glass theory, and corrections again come only from subgraphs that are free of dangling edges, provided that the auxiliary probability distributions of the expansion are chosen to be a fixed point of the survey-propagation equation. The same loop series expansion can be performed for higher-level partition functions, obtaining the higher-level RSB BP free-energy functionals (and the correction terms) and message-passing equations without using the BP approximation.
机译：对图上的一般统计模型的分割函数进行循环级数展开。如果将展开的辅助概率分布选择为置信传播方程的固定点，则循环序列的第一项将在均值的副本对称水平上赋予BethePeierls（BP）自由能泛函自旋玻璃理论和更正仅由没有悬空边缘的子图提供。该结果概括了Chertkov和Chernyak在二进制统计模型上的早期工作。如果信念传播方程具有多个固定点，则对大分割函数执行循环级数展开。该系列的第一项在平均场自旋玻璃理论的第一步复制-对称破坏（RSB）级别上提供了BP自由能函数，并且校正仅再次来自没有悬挂边缘的子图，前提是将展开的辅助概率分布选择为调查传播方程的不动点。可以对更高级别的分区函数执行相同的循环序列扩展，无需使用BP近似即可获得更高级别的RSB BP自由能函数（和校正项）和消息传递方程式。

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