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We consider the problem of characterizing Bayesian networks up tounconditional equivalence, i.e., when directed acyclic graphs (DAGs) have thesame set of unconditional $d$-separation statements. Each unconditionalequivalence class (UEC) is uniquely represented with an undirected graph whoseclique structure encodes the members of the class. Via this structure, weprovide a transformational characterization of unconditional equivalence; i.e.,we show that two DAGs are in the same UEC if and only if one can be transformedinto the other via a finite sequence of specified moves. We also extend thischaracterization to the essential graphs representing the Markov equivalenceclasses (MECs) in the UEC. UECs partition the space of MECs and are easilyestimable from marginal independence tests. Thus, a characterization ofunconditional equivalence has applications in methods that involve searchingthe space of MECs of Bayesian networks.

A Transformational Characterization of Unconditionally Equivalent Bayesian Networks