一般图形模型的配分函数回路系列：自由能修正和信息传递方程
Partition function loop series for a general graphical model: free energy corrections and message-passing equations
摘要：
对一般统计模型的配分函数在图上进行了圈级数展开。如果扩展的辅助概率分布被选择为置信传播方程的不动点，则循环级数的第一项给出平均场自旋玻璃理论的复制对称水平上的贝特-佩尔斯自由能泛函，并且校正仅由没有悬挂边的子图贡献。这个结果推广了Chertkov和Chernyak关于二元统计模型的早期工作。如果置信传播方程具有多个不动点，则对大配分函数执行循环级数展开。这个级数的第一项给出了平均场自旋玻璃理论第一步复制对称破缺（RSB）水平上的贝特-佩尔斯自由能泛函，而修正同样只来自没有悬挂边的子图，前提是扩展的辅助概率分布被选为巡天传播方程的不动点。对于更高级别的配分函数，可以执行相同的循环级数展开，从而获得更高级别的RSB Bethe-Peierls自由能泛函（和校正项）和消息传递方程，而不使用Bethe-Peierls近似。
英文摘要：
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 Bethe-Peierls 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 generalize 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 Bethe-Peierls 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 Bethe-Peierls free energy functionals (and the correction terms) and message-passing equations without using the Bethe-Peierls approximation.
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