In this paper we give a complete characterization of the equivalence classes of CEGs or equivalently of staged trees. This model class cannot be unambiguously indexed by its graphical properties. However, we are able to show that a polynomial defined on an underlying graph codes all relevant characteristics and is common to all representations of the same model. Furthermore, simple transformations on that polynomial enable us to traverse the statistical equivalence class of these graphs. So one can design efficient algorithms over these classes. We illustrate our results throughout the paper, finishing with a real analysis of the implicit dependence relationships found in a dataset.
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