A Polynomial Time Algorithm For Determining DAG Equivalence in the Presence of Latent Variables and Selection Bias 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 Collections: Mathematics