Markov equivalence of marginalized local independence graphs

By Søren Wengel Mogensen, Niels Richard Hansen. 2018

50 citations recorded by the source corpus.

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 $\mu$-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. For statistical applications, it is pivotal to characterize graphs that induce the same independence relations as such a Markov equivalence class of graphs is the object that is ultimately identifiable from observational data. Our main result is that for directed mixed graphs with $\mu$-separation each Markov 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 edge markings. This graph encodes all the information about the edges that is identifiable from the independence relations, and furthermore it can be computed efficiently from the maximal graph.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Signal Processing

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Citations in the corpus

Causal Models for Dynamical Systems (2020, background): “on seems less justied in the case of dynamical processes. Also, considering local independence and assuming causal suciency, Markov equivalence classes contain only a single directed acyclic graph [Mogensen and Hansen, 2019]. Both of these points may prove to be useful for causal discovery. Furtherm”
Foundations of structural causal models with cycles and latent variables (2016, background): “However, causal cycles may arise when one approximates such systems over time [16, 39, 40] or when one describes the equilibrium states of these systems [3, 5, 24, 26, 30, 43, 53].”
Causal Learning for Partially Observed Stochastic Dynamical Systems (2018, methodology): “Using properties of maximal DMGs, Mogensen and Hansen (2018) showed how one can construct the DMEG efficiently.”