近似Bethe配分函数
Approximating the Bethe partition function
摘要：
当置信传播（BP）收敛时，它会收敛到贝特自由能$F$的一个稳定点，并且通常非常精确。然而，它可能只收敛到局部最优值，也可能根本不收敛。最近引入了一种用于吸引二进制成对MRF的算法，该算法保证在多项式时间内返回$F$的全局最小值的$ε$ -近似，提供最大度$Δ=O(\log n)$，其中$n$是变量的数量。在这里，我们显着改善这种算法，并得出几个结果，包括一个新的方法的基础上分析一阶导数的$F$，这导致性能通常是远远优于产生一个完全多项式时间近似方案（FPTAS）的吸引力模型没有任何程度的限制。此外，该方法适用于一般（非吸引力）的模型，虽然在这种情况下没有多项式时间保证，导致的重要结果，近似$\log$的Bethe分区函数，$\log Z_B=-\min F$，为一般模型添加$ε$ -准确性可以减少到一个离散MAP推理问题。我们探讨了应用预测城市电网上的设备故障，并证明了Bethe近似可以很好地执行，即使BP无法收敛。
英文摘要：
When belief propagation (BP) converges, it does so to a stationary point of the Bethe free energy $F$, and is often strikingly accurate. However, it may converge only to a local optimum or may not converge at all. An algorithm was recently introduced for attractive binary pairwise MRFs which is guaranteed to return an $ε$-approximation to the global minimum of $F$ in polynomial time provided the maximum degree $Δ=O(\log n)$, where $n$ is the number of variables. Here we significantly improve this algorithm and derive several results including a new approach based on analyzing first derivatives of $F$, which leads to performance that is typically far superior and yields a fully polynomial-time approximation scheme (FPTAS) for attractive models without any degree restriction. Further, the method applies to general (non-attractive) models, though with no polynomial time guarantee in this case, leading to the important result that approximating $\log$ of the Bethe partition function, $\log Z_B=-\min F$, for a general model to additive $ε$-accuracy may be reduced to a discrete MAP inference problem. We explore an application to predicting equipment failure on an urban power network and demonstrate that the Bethe approximation can perform well even when BP fails to converge.
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