Classical graphical modeling of multivariate random vectors uses graphs to encode conditional independence. In graphical modeling of multivariate stochastic processes, graphs may encode so-called local independence analogously. If some coordinate processes of the multivariate stochastic process are unobserved, the local independence graph of the observed coordinate processes is a directed mixed graph (DMG). Two DMGs may encode the same local independences in which case we say that they are Markov equivalent. Markov equivalence is a central notion in graphical modeling. We show that deciding Markov equivalence of DMGs is coNP-complete, even under a sparsity assumption. As a remedy, we introduce a collection of equivalence relations on DMGs that are all less granular than Markov equivalence and we say that they are weak equivalence relations. This leads to feasible algorithms for naturally occurring computational problems related to weak equivalence of DMGs. The equivalence classes of a weak equivalence relation have attractive properties. In particular, each equivalence class has a greatest element which leads to a concise representation of an equivalence class. Moreover, these equivalence relations define a hierarchy of granularity in the graphical modeling which leads to simple and interpretable connections between equivalence relations corresponding to different levels of granularity.

翻译：经典的多变量随机向量图模型使用图来编码条件独立性。在多变量随机过程的图模型中，图可以类比地编码所谓的局部独立性。若多变量随机过程中的某些坐标过程未被观测，则观测坐标过程的局部独立图是一个有向混合图（DMG）。两个有向混合图可能编码相同的局部独立性，此时我们称它们为马尔可夫等价的。马尔可夫等价是图模型中的核心概念。我们证明，即使在稀疏性假设下，判定有向混合图的马尔可夫等价性也是coNP完全的。为此，我们引入了一组关于有向混合图的等价关系集合，这些关系均比马尔可夫等价性更粗粒度，我们称它们为弱等价关系。这为与有向混合图弱等价相关的自然计算问题提供了可行的算法。弱等价关系的等价类具有良好的性质，特别地，每个等价类存在一个最大元素，从而可简洁地表示等价类。此外，这些等价关系在图模型中定义了一个粒度层次结构，使得不同粒度层次对应的等价关系之间具有简单且可解释的关联。