Duality between faithfulness assumptions in Graphical models

In this paper we analyze the duality between two faithfulness assumptions that can be defined on a given multivariate probability distribution of a set of random variables. The first pertains to faithfulness to its concentration graph and the second pertains to faithfulness to its covariance graph. The vertices in both these graphs are in a oneto-one correspondence with the set of variables in the random vector. The concentration graph is an undirected graph constructed by looking through conditional independences between each pair of variables given the remaining variables and the covariance graph is constructed by looking through marginal independences between each pair of variables. The absence of an edge in the graph corresponds to conditional or marginal independences respectively. On each graph a separation criteria is defined which implies conditional independences present in the probability distribution: this is termed the Global Markov property. Furthermore, the faithfulness assumption is said to be satisfied when all the independence conditional statements in the probability distributions are represented in the graph. In this paper we analyze the duality between these two faithfulness hypothesis. We also prove that when the both assumptions are simultaneously satisfied, i.e., the bi-faithfulness property, all the connected components in the concentration and in the covariance graphs are either complete or all their separators have a cardinality equal to |V |− 2 where V is the number of variables.