Implications of Faithfulness in Graphical Models

Concentration graph models and covariance graph models are two of the widely studied classes of graphical models. They are specified through pairwise relationships between variables. Under suitable conditions, they can be used to read conditional independence relations at the level of sets of variables. It’s known that faithfulness property is filled when the graph allows identifying the whole set condition independence statements. This paper studies the implications of imposing the faithfulness assumption on either the covariance or concentration graphs. We demonstrate that if a probability distribution is faithful to its concentration graph. The corresponding covariance graph is a union of complete connected components, i.e., each connected component cannot have any marginal independence among its nodes. We also prove a dual result when the distribution is faithful to its covariance graph. The general implications of the results are far-reaching. First, the result formalizes the long-held notion in the graphical models’ community that faithfulness is a very restrictive assumption. Second, we show that estimation procedures in graphical models by low-order conditioning may lead to erroneous conclusions. Since these procedures effectively search for models in a very restrictive class of probability. distributions.