Description of structures of stochastic conditional independence by means of faces and imsets .2. Basic theory
The central concept of face (with respect to an arbitrary continuous linear ordering on imsets) and the corresponding deductive mechanism (facial implication) are introduced. The class of all faces constitutes a lattice which, in the case of finitely established orderings, is shown to be finite. Moreover, its atoms and co-atoms are characterized and two possible representations of faces are treated: by means of generating imsets and by means of portraits. Then a concrete ordering, called the structural ordering, is studied. Faces with respect to this ordering are identified with a class of dependency models including all models of probabilistic conditional independence structures.
conditional independence face imset structural ordering structural semigraphoid