AMP链图：最小分离子与结构学习算法
AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms
摘要：
我们解决的问题，找到一个最小的分离器中的Anderson-Madigan-Perlman链图（AMP CG），即，找到一组Z的节点，分离一个给定的不相邻的节点对，使没有真子集Z分离该对。我们分析了这个问题的几个版本，并为每个版本提供多项式时间算法。这些包括从一个有限的节点集合中找到一个最小分隔符，为两个给定的不相交集合找到一个最小分隔符，以及测试一个给定的分隔符是否是最小的。为了解决从数据中学习AMP CG结构的问题，我们证明了PC类算法（Pena，2012）是依赖于顺序的，在这个意义上，输出可以取决于变量给定的顺序。我们提出了几个修改的PC类算法，删除部分或全部的顺序依赖。我们还扩展了（Xie et al.，2006）学习AMP CG，其中包括BN作为一个特殊的情况下，在忠诚的假设。我们证明了我们的扩展使用最小分离器的结果的正确性。在我们的实验中使用标准的基准测试和综合生成的模型和数据证明了我们的基于分解的方法，称为LCD-AMP，与（修改版本）PC类算法相比，具有竞争力的性能。LCD-AMP算法通常优于PC类算法，我们对PC类算法的修改比原始PC类算法学习更类似于底层地面真值图的结构，特别是在高维设置中。特别是，我们的经验表明，这两种算法的结果是更准确和稳定的样本量是相当大的，底层的图形是稀疏的。
英文摘要：
We address the problem of finding a minimal separator in an Andersson-Madigan-Perlman chain graph (AMP CG), namely, finding a set Z of nodes that separates a given nonadjacent pair of nodes such that no proper subset of Z separates that pair. We analyze several versions of this problem and offer polynomial-time algorithms for each. These include finding a minimal separator from a restricted set of nodes, finding a minimal separator for two given disjoint sets, and testing whether a given separator is minimal. To address the problem of learning the structure of AMP CGs from data, we show that the PC-like algorithm (Pena, 2012) is order-dependent, in the sense that the output can depend on the order in which the variables are given. We propose several modifications of the PC-like algorithm that remove part or all of this order-dependence. We also extend the decomposition-based approach for learning Bayesian networks (BNs) proposed by (Xie et al., 2006) to learn AMP CGs, which include BNs as a special case, under the faithfulness assumption. We prove the correctness of our extension using the minimal separator results. Using standard benchmarks and synthetically generated models and data in our experiments demonstrate the competitive performance of our decomposition-based method, called LCD-AMP, in comparison with the (modified versions of) PC-like algorithm. The LCD-AMP algorithm usually outperforms the PC-like algorithm, and our modifications of the PC-like algorithm learn structures that are more similar to the underlying ground truth graphs than the original PC-like algorithm, especially in high-dimensional settings. In particular, we empirically show that the results of both algorithms are more accurate and stabler when the sample size is reasonably large and the underlying graph is sparse.
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