Markov equivalence of marginalized local independence graphs

Symmetric independence relations are often studied using graphical representations. Ancestral graphs or acyclic directed mixed graphs with m-separation provide classes of symmetric graphical independence models that are closed under marginalization. Asymmetric independence relations appear naturally for multivariate stochastic processes, for instance, in terms of local independence. However, no class of graphs representing such asymmetric independence relations, which is also closed under marginalization, has been developed. We develop the theory of directed mixed graphs with . -separation and show that this provides a graphical independence model class which is closed under marginalization and which generalizes previously considered graphical representations of local independence. Several graphs may encode the same set of independence relations and this means that in many cases only an equivalence class of graphs can be identified from observational data. For statistical applications, it is therefore pivotal to characterize graphs that induce the same independence relations. Our main result is that for directed mixed graphs with . -separation each equivalence class contains a maximal element which can be constructed from the independence relations alone. Moreover, we introduce the directed mixed equivalence graph as the maximal graph with dashed and solid edges. This graph encodes all information about the edges that is identifiable from the independence relations, and furthermore it can be computed efficiently from the maximal graph.