JdS2012


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Résumé 31 :

Pseudographoid axioms and the covariance global Markovproperty
malouche, dhafer ; rajaratnam, bala
ESSAI-U2S-ENIT

The covariance and the concentration graphs are two undirected graphs associated with a random vector $X$. Each vertex in these graphs corresponds to a variable in the vector $X$. The absence of an edge in the covariance graph between any pair of vertices encodes pairwise marginal independence between the variables represented by these vertices. The absence of an edge between any pair of vertices in the concentration graph encodes conditional independence between these two variables given the rest. More complex conditional independence relationships at the level of sets of variables can also be deduced from separation statements from these two graphs by using the global Markov property with respect to the graph. The equivalence of the pairwise and global Markov property for concentration graphs is valid under a general condition called the pseudographoid axiom. The analogous result for the covariance global Markov property however requires the verification of a longer list of properties. One typical set of conditions are the so-called weak transitivity composition(WTC) graphoid axioms. Hence the global Markov property cannot be invoked as easily for covariance graphs. The aim of this paper is to prove that the covariance global Markov property can be established under simpler conditions, in particular when the converse of the pseudographoid axiom is satisfied. The result therefore significantly reduces the conditions that need to verified for the application of the covariance global Markov property, and in the process establishes a duality between the covariance and concentration global Markov properties.