Characterizing Distribution Equivalence for Cyclic and Acyclic Directed Graphs
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The main way for defining equivalence among acyclic directed graphs is based on the conditional independencies of the distributions that they can generate. However, it is known that when cycles are allowed in the structure, conditional independence is not a suitable notion for equivalence of two structures, as it does not reflect all the information in the distribution that can be used for identification of the underlying structure. In this paper, we present a general, unified notion of equivalence for linear Gaussian directed graphs. Our proposed definition for equivalence is based on the set of distributions that the structure is able to generate. We take a first step towards devising methods for characterizing the equivalence of two structures, which may be cyclic or acyclic. Additionally, we propose a score-based method for learning the structure from observational data.
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