 
Summary: 1
A Polynomial Time Algorithm For Determining DAG Equivalence in the
Presence of Latent Variables and Selection Bias
by Peter Spirtes (Department of Philosophy, Carnegie Mellon University, ps7z@andrew.cmu.edu)
and Thomas Richardson (Department of Statistics, University of Washington)
Following the terminology of Lauritzen et. al. (1990) say that a probability measure over a set of variables
V satisfies the local directed Markov property for a directed acyclic graph (DAG) G with vertices V
if and only if for every W in V, W is independent of the set of all its nondescendants conditional on the set
of its parents. One natural question that arises with respect to DAGs is when two DAGs are ``statistically
equivalent''. One interesting sense of ``statistical equivalence'' is ``dseparation equivalence'' (explained in
more detail below.) In the case of DAGs, dseparation equivalence is also corresponds to a variety of other
natural senses of statistical equivalence (such as representing the same set of distributions). Theorems
characterizing dseparation equivalence for directed acyclic graphs and that can be used as the basis for
polynomial time algorithms for checking dseparation equivalence were provided by Verma and Pearl
(1990), and Frydenberg (1990). The question we will examine is how to extend these results to cases where
a DAG may have latent (unmeasured) variables or selection bias (i.e. some of the variables in the DAG
have been conditioned on.) Dseparation equivalence is of interest in part because there are algorithms for
constructing DAGs with latent variables and selection bias that are based on observed conditional
independence relations. For this class of algorithms, it is impossible to determine which of two dseparation
equivalent causal structures generated a given probability distribution, given only the set of conditional
