  In this paper, we derive optimality results for greedy Bayesian-network
search algorithms that perform single-edge modifications at each step and use
asymptotically consistent scoring criteria. Our results extend those of Meek
(1997) and Chickering (2002), who demonstrate that in the limit of large
datasets, if the generative distribution is perfect with respect to a DAG
defined over the observable variables, such search algorithms will identify
this optimal (i.e. generative) DAG model. We relax their assumption about the
generative distribution, and assume only that this distribution satisfies the
{em composition property} over the observable variables, which is a more
realistic assumption for real domains. Under this assumption, we guarantee that
the search algorithms identify an {em inclusion-optimal} model; that is, a
model that (1) contains the generative distribution and (2) has no sub-model
that contains this distribution. In addition, we show that the composition
property is guaranteed to hold whenever the dependence relationships in the
generative distribution can be characterized by paths between singleton
elements in some generative graphical model (e.g. a DAG, a chain graph, or a
Markov network) even when the generative model includes unobserved variables,
and even when the observed data is subject to selection bias.
