Greedy Causal Discovery is Geometric
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Finding a directed acyclic graph (DAG) that best encodes the conditional independence statements observable from data is a central question within causality. Algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful, for example the GES, GIES, and MMHC algorithms. In 2010, Studený, Hemmecke and Lindner introduced the characteristic imset polytope, CIM_p, whose vertices correspond to Markov equivalence classes, as a way of transforming causal discovery into a linear optimization problem. We show that the moves of the aforementioned algorithms are included within classes of edges of CIM_p and that restrictions placed on the skeleton of the candidate DAGs correspond to faces of CIM_p. Thus, we observe that GES, GIES, and MMHC all have geometric realizations as greedy edge-walks along CIM_p. Furthermore, the identified edges of CIM_p strictly generalize the moves of these algorithms. Exploiting this generalization, we introduce a greedy simplex-type algorithm called greedy CIM, and a hybrid variant, skeletal greedy CIM, that outperforms current competitors among hybrid and constraint-based algorithms.
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