On characterizing Inclusion of Bayesian Networks
  Every directed acyclic graph (DAG) over a finite non-empty set of variables
(= nodes) N induces an independence model over N, which is a list of
conditional independence statements over N.The inclusion problem is how to
characterize (in graphical terms) whether all independence statements in the
model induced by a DAG K are in the model induced by a second DAG L. Meek
(1997) conjectured that this inclusion holds iff there exists a sequence of
DAGs from L to K such that only certain 'legal' arrow reversal and 'legal'
arrow adding operations are performed to get the next DAG in the sequence.In
this paper we give several characterizations of inclusion of DAG models and
verify Meek's conjecture in the case that the DAGs K and L differ in at most
one adjacency. As a warming up a rigorous proof of well-known graphical
characterizations of equivalence of DAGs, which is a highly related problem, is
given.
