A graphical characterization of lattice conditional independence models

Authors:
Steen A. Andersson;David Madigan;Michael D. Perlman;Christopher M. Triggs
Affiliations:
Indiana University, University of Washington and University of Auckland;Indiana University, University of Washington and University of Auckland;Indiana University, University of Washington and University of Auckland;Indiana University, University of Washington and University of Auckland
Venue:
Annals of Mathematics and Artificial Intelligence
Year:
1997

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Abstract

Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of non‐monotone missing data patterns and of nonnested dependent linear regression models (\equiv seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graph‐theoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (possibly superexponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities.