对Bethe近似和Bethe方法的新认识
New Understanding of the Bethe Approximation and the Replica Method
摘要：
在这篇论文中，我们对Bethe近似提出了新的推广，并对复制方法有了新的理解。Bethe近似是图模型的一种有效近似，它给出了许多图模型的配分函数的渐近精确估计。Bethe近似解释了著名的消息传递算法，置信传播，这是精确的树图形模型。它也是已知的集群变分方法给出了广义贝特近似，称为菊池近似，产生广义信念传播。本文给出了Bethe逼近的一系列新的推广，称之为渐近Bethe逼近。渐近Bethe近似是从最近由Vontobel得到的Bethe自由能的图覆盖的特征中推导出来的。利用Watanabe和Fukumizu关于Bethe熵的Hessian的结果，可以用边缘zeta函数来表示渐近Bethe逼近。证明了渐近Bethe逼近在一定条件下优于传统的Bethe逼近。为此，Chertkov和Chernyak的循环演算公式，这表明Bethe近似的误差可以表示为对应于广义循环的权重和，并推广到非二进制有限字母表，通过使用信息几何的概念。
英文摘要：
In this thesis, new generalizations of the Bethe approximation and new understanding of the replica method are proposed. The Bethe approximation is an efficient approximation for graphical models, which gives an asymptotically accurate estimate of the partition function for many graphical models. The Bethe approximation explains the well-known message passing algorithm, belief propagation, which is exact for tree graphical models. It is also known that the cluster variational method gives the generalized Bethe approximation, called the Kikuchi approximation, yielding the generalized belief propagation. In the thesis, a new series of generalization of the Bethe approximation is proposed, which is named the asymptotic Bethe approximation. The asymptotic Bethe approximation is derived from the characterization of the Bethe free energy using graph covers, which was recently obtained by Vontobel. The asymptotic Bethe approximation can be expressed in terms of the edge zeta function by using Watanabe and Fukumizu's result about the Hessian of the Bethe entropy. The asymptotic Bethe approximation is confirmed to be better than the conventional Bethe approximation on some conditions. For this purpose, Chertkov and Chernyak's loop calculus formula is employed, which shows that the error of the Bethe approximation can be expressed as a sum of weights corresponding to generalized loops, and generalized for non-binary finite alphabets by using concepts of information geometry.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
