Faithfulness and learning hypergraphs from discrete distributions

The concepts of faithfulness and strong-faithfulness are important for statistical learning of graphical models. Graphs are not sufficient for describing the association structure of a discrete distribution. Hypergraphs representing hierarchical log-linear models are considered instead, and the concept of parametric (strong-)faithfulness with respect to a hypergraph is introduced. The strength of association in a discrete distribution can be quantified with various measures, leading to different concepts of strong-faithfulness. It is proven that strong-faithfulness defined in terms of interaction parameters ensures the existence of uniformly consistent parameter estimators and enables building uniformly consistent procedures for a hypergraph search. Lower and upper bounds for the proportions of distributions that do not satisfy strong-faithfulness are computed for different parameterizations and measures of association.