Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find all fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.

中文翻译：

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信念传播的不动点—基于多项式同伦连续性的分析

置信传播（BP）是对任意图形模型执行近似推断的一种迭代方法。BP是否收敛以及解是否是唯一的固定点取决于模型的结构和参数化。要了解这种依赖性，有趣的是找到 全部 定点。在这项工作中，我们制定了一组多项式方程，其解对应于BP不动点。为了解决这种非线性系统，我们提出了数值多项式同伦连续（NPHC）方法。在二进制Ising模型和纠错代码上进行的实验表明，我们的方法如何能够获得所有BP固定点。在具有固定参数的Ising模型中，我们展示了结构如何影响固定点的数量和收敛性。我们进一步评估了边际及其加权组合的准确性。加权边际及其各自的分区函数可提高所有实验的准确性。与BP不动点的唯一性暗示收敛的猜想相反，我们发现即使存在唯一不动点，BP也无法收敛的图。