A new notion of convexity in digraphs with an application to Bayesian networks

We introduce a new notion of convexity in digraphs, which we call incoming-path convexity, and prove that the incoming-path convexity space of a digraph is a convex geometry (that is, it satisfies the Minkowski–Krein–Milman property) if and only if the digraph is acyclic. Moreover, we prove that incoming-path convexity is adequate to characterize collapsibility of models generated by Bayesian networks. Based on these results, we also provide simple linear algorithms to solve two topical problems on Markov properties of a Bayesian network (that is, on conditional independences valid in a Bayesian network).

[1]  Marina Moscarini,et al.  Computing simple-path convex hulls in hypergraphs , 2011, Inf. Process. Lett..

[2]  John L. Pfaltz,et al.  Closure lattices , 1996, Discret. Math..

[3]  Dan Geiger,et al.  Identifying independence in bayesian networks , 1990, Networks.

[4]  Marina Moscarini,et al.  Decomposability of Abstract and Path-Induced Convexities in Hypergraphs , 2015, Discuss. Math. Graph Theory.

[5]  Marina Moscarini,et al.  Equivalence between Hypergraph Convexities , 2011 .

[6]  Francesco M. Malvestuto.  Testing implication of hierarchical log-linear models for probability distributions , 1996, Stat. Comput..

[7]  Francesco M. Malvestuto.  Canonical and monophonic convexities in hypergraphs , 2009, Discret. Math..

[8]  John L. Pfaltz,et al.  Closure systems and their structure , 2001, Inf. Sci..

[9]  Martin G. Everett,et al.  The hull number of a graph , 1985, Discret. Math..

[10]  Elitza N. Maneva,et al.  Pruning processes and a new characterization of convex geometries , 2007, Discret. Math..

[11]  J. Pfaltz.  The Role of Continuous Processes in Cognitive Development , 2015 .

[12]  Francesco M. Malvestuto,et al.  A hypergraph-theoretic analysis of collapsibility and decomposability for extended log-linear models , 2001, Stat. Comput..

[13]  Paul H. Edelman,et al.  The theory of convex geometries , 1985 .

[14]  M. Farber,et al.  Convexity in graphs and hypergraphs , 1986 .

[15]  Steffen L. Lauritzen,et al.  Independence properties of directed markov fields , 1990, Networks.

[16]  J. Pfaltz.  Convexity in directed graphs , 1971 .

[17]  John L. Pfaltz,et al.  Closure spaces that are not uniquely generated , 2005, Discret. Appl. Math..