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

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 Win V, Wis independent of the set of all its non-descendants 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 "conditional independence equivalence" which holds when two DAGs entail the same set of conditional independence relations. In the case of DAGs, conditional independence equivalence also corresponds to a variety of other natural senses of statistical equivalence (such as representing the same set of distributions). Theorems characterizing conditional independence equivalence for directed acyclic graphs and that can be used as the basis for polynomial time algorithms for checking conditional independence 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.) Conditional independence 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 conditional independence equivalent causal structures generated a given probability distribution, given only the set of conditional independence and dependence relations true of the observed distribution. We will describe a polynomial (in the number of vertices) time algorithm for determining when two DAGs which may have latent variables or selection bias are conditional independence equivalent. A DAG G entails a conditional independence relation R if and only if R is true in every probability measure satisfying the local directed Markov property for G. (We place definitions and sets of variables in boldface.) Pearl, Geiger, and Verma (Pearl 1988) have shown that there is a graphical relation, ct-separation, that holds among three disjoint sets of variable A, and B, and C in DAG G if and only if G entails that A is independent of B given C. A vertex Y is a collider on an undirected path U if U contains a subpath X ➔ Y ~ Z. Say that a vertex V on an undirected path U between X and Y is active on U given Z (Z not containing X and Y) if and only if either V is not a collider on U and not in Z, or V is a collider on U and is an ancestor of Z. For three disjoint sets of variables A, B, and C, A is d-connected to B given C in graph G, if and only if there is an undirected path from some member of A to a member of B such that every vertex on U is active given C; for three disjoint sets of variables A, B, and C, A is dseparated from B given C in graph G, if and only A is not cl-connected to B given C. Two DAGs are conditional independence equivalent if and only if they have the same vertices and entail the same set of conditional independence relations . If two DAGs G 1 and G2 are conditional independence equivalent, the set of distributions that satisfy the local directed Markov property for G1 equals the set of distribution that satisfy the local directed Markov property for G2• Theorems that provide the basis for polynomial time algorithms for testing conditional independence equivalence for DAGs were given in Verma and Pearl (1990), for cyclic directed graphs in Richardson (1994), and for directed acyclic graphs with latent variables in Spirtes and Verma (1992).