The Bethe free energy approximation provides an effective way for relaxing NP-hard problems of probabilistic inference. However, its accuracy depends on the model parameters and particularly degrades if a phase transition in the model occurs. In this work, we analyze when the Bethe approximation is reliable and how this can be verified. We argue and show by experiment that it is mostly accurate if it is convex on a submanifold of its domain, the 'Bethe box'. For verifying its convexity, we derive two sufficient conditions that are based on the definiteness properties of the Bethe Hessian matrix: the first uses the concept of diagonal dominance, and the second decomposes the Bethe Hessian matrix into a sum of sparse matrices and characterizes the definiteness properties of the individual matrices in that sum. These theoretical results provide a simple way to estimate the critical phase transition temperature of a model. As a practical contribution we propose $\texttt{BETHE-MIN}$, a projected quasi-Newton method to efficiently find a minimum of the Bethe free energy.

翻译：Bethe自由能近似为概率推断中的NP难问题提供了一种有效的松弛方法。然而，其精度取决于模型参数，当模型发生相变时精度会显著下降。本文分析了Bethe近似在何种条件下可靠以及如何验证其可靠性。我们通过理论论证与实验表明：若该近似在其定义域子流形（即“Bethe盒”）上具有凸性，则其通常能保持较高精度。为验证凸性，我们基于Bethe Hessian矩阵的正定性推导出两个充分条件：第一个条件利用对角占优概念，第二个条件将Bethe Hessian矩阵分解为稀疏矩阵之和，并通过分析各分矩阵的正定性来判定整体性质。这些理论结果为估计模型的临界相变温度提供了简明方法。作为实际贡献，我们提出$\texttt{BETHE-MIN}$方法——一种投影拟牛顿算法，可高效求解Bethe自由能的极小值。