Graphical models use graphs to represent conditional independence structure in the distribution of a random vector. In stochastic processes, graphs may represent so-called local independence or conditional Granger causality. Under some regularity conditions, a local independence graph implies a set of independences using a graphical criterion known as $\delta$-separation, or using its generalization, $\mu$-separation. This is a stochastic process analogue of $d$-separation in DAGs. However, there may be more independences than implied by this graph and this is a violation of so-called faithfulness. We characterize faithfulness in local independence graphs and give a method to construct a faithful graph from any local independence model such that the output equals the true graph when Markov and faithfulness assumptions hold. We discuss various assumptions that are weaker than faithfulness, and we explore different structure learning algorithms and their properties under varying assumptions.

翻译：图形模型使用图来表示随机向量分布中的条件独立结构。在随机过程中，图可以表示所谓的局部独立性或条件格兰杰因果关系。在一定的正则条件下，局部独立性图通过一种称为$\delta$-分离的图形准则，或其推广形式$\mu$-分离，来推导出一组独立性关系。这是有向无环图中$d$-分离在随机过程中的类比。然而，实际独立性可能多于该图所隐含的独立性，这违反了所谓的忠实性假设。我们刻画了局部独立性图中的忠实性，并给出了一种从任意局部独立性模型构建忠实图的方法，使得当马尔可夫性和忠实性假设成立时，输出的图与真实图一致。我们讨论了比忠实性更弱的各种假设，并探讨了不同结构学习算法及其在不同假设下的性质。