t-separation and d-separation for Directed Acyclic Graphs

Transcription

1 t-separation and d-separation for Directed Acyclic Graphs Yanming Di Technical Report No. 552 Department of Statistics, University of Washington, Seattle, WA February 10, 2009 Abstract In recent work, Sullivant and Talaska [2008] introduced a new separation criterion, t-separation, for DAGs. They showed that for a Gaussian DAG model, t-separation characterizes when a submatrix of the variance-covariance matrix will have non-full rank. In particular, special forms of t-separation characterize all conditional independence statements for the Gaussian DAG model. This indirectly proved that these special forms of t-separation are equivalent to the d-separation criterion, since d-separation also characterizes all conditional independence statements for a Gaussian DAG model. In this report, we present a direct graph-theoretic proof of the equivalence between the special forms of t-separation and d-separation. 1

2 1 Introduction A DAG (directed acyclic graph) is a graph with only directed edges and without directed cycles. Let G = (V, E) be a graph with vertex set V and edge set E. A sequence of edges for which the corresponding vertices contains no repetitions is called a (simple) path. A path between vertices a and b in a DAG is said to be d-connecting given the vertex set C if 1. every non-collider on the path is in C, and 2. every collider on the path is in Z or has a descendant in Z. If there is no d-connecting path between a and b, then a and b are said to be d-separated. For subsets A, and C of vertices, we say C d-separates A from, if for every pair (a, b), a A and b, a, b are d-separated given C. For multivariate distributions that factorize according to a DAG, the global Markov property associates conditional independence relations holding for the distributions with the d-separation properties of the DAG (see Lauritzen [1996]). For a directed path G from i to j, i j, we call i the source of the path and j the sink of the path. A trek T ij in G from i to j consists of a directed path P ki from k to i and a directed path P kj from k to j. Vertices in P ki are said to be on the i-side of the trek and vertices in P kj are on the j-side. The common source k is called the top of the trek. In this definition of a trek, one or both of P ki and P kj can be a single vertex, with no edge. Note that k is on both sides of the trek. The trek T ij is simple, if it contains no repetitive vertices. In recent work, Sullivant and Talaska [2008] introduced a new separation criterion, t-separation, for DAGs: Definition 1.1 (Definition 2.7 of Sullivant and Talaska [2008]). Let A,, C A and C be four subsets of V(G) which need not be disjoint. The pair (C A, C ) trek separates (or t-separates) A from if every trek from A to passes through either a vertex on the A-side of the trek, or a vertex in C on the -side of the trek. Sullivant and Talaska [2008] showed that for Gaussian random variables X V whose distribution factorizes according to the DAG G, this t-separation criterion fully characterizes when a sub-matrix of the variance-covariance matrix of X V will have reduced rank (see Theorem 2.8 in Sullivant and 2

3 Talaska [2008]). In particular, t-separation characterizes all conditional independence statements for Gaussian DAG models: Theorem 1.2 (Theorem 2.11 of Sullivant and Talaska [2008]). The conditional independence statement X A X X C holds for (all Gaussian distributions that factorize according to) the graph G if and only if there is a partition C A C = C of C such that (C A, C ) t-separations A C and C in G. Since the d-separation criterion also characterizes all the conditional independence statements. The special forms of t-separations in Theorem 1.2 must be equivalent to corresponding d-separations. Sullivant and Talaska [2008] pointed this out as a corollary: Corollary 1.3 (Corollary 2.12 of Sullivant and Talaska [2008]). (For disjoint vertex sets A,, C.) A set C d-separates A from in G if and only if there is a partition C = C A C such that {C A, C } t-separates A C and C. In this report, we present a direct proof to this corollary. The proof uses only structural properties of graphs. As a by-product, when C d-separates A and, the proof provides an algorithm for partitioning C into C A and C so that (C A, C ) will t-separates A C and C. Since t-separation and d-separation are equivalent, the statement in Theorem 1.2 is true for all distributions (not just Gaussian distributions) that factorizes according to the DAG G: Corollary 1.4. If there is a partition C A C = C of C such that (C A, C ) t-separations A C and C in G, then the conditional independence statement X A X X C holds for all distributions that factorize according to the graph G. 2 The proof. Proof. t-separation implies d-separation: Suppose (C A, C ) t-separates A C and C. Assume there is a simple path P ab d-connecting a vertex a A and a vertex b given C, then no non-collider on the path is in C and each collider is in C or has a descendant in C. If there is no collider on the path, then the path is a trek between a and b. y t-separation, the path(trek) passes through either a vertex in C A or a vertex in C. This contradicts that no 3

4 non-collider on the path is in C. If there are colliders on the path, let c 1, c 2,...,c n be the list of colliders in the order of appearance along the path from a to b. These colliders divide path P A into sub-paths T ac1, T c1 c 2,...,T cnb, each sub-path being a trek. For each collider c i, let d i = c i if c i C; otherwise, there is a directed path from c i to a vertex in C, let d i be the first vertex in C on this directed path. Let P ci d i be the portion of the directed path between c i and d i. d i is the only vertex on P ci d i that is in C. I will show by induction that d i C for all i. The path resulting from concatenating the trek T ac1 and the directed path P c1 d 1 is a trek (not necessarily simple) connecting a vertex a in A C and a vertex d 1 in C. There is an arrow head at d 1 and so d 1 is on only the C side of this trek. No vertex on the A C side of this trek is in C and d 1 is the only vertex on the C side that is in C, so d 1 is in C by t-separation. Suppose d i 1 C. Concatenating P ci 1 d i 1, T ci 1 c i and P ci d i gives a trek connecting d i 1 C A C and d i C C. No vertex on the A C side of this trek is in C A and d i is the only vertex on the C side that is in C, so d i has to be in C by t-separation. y induction, we conclude that all d i have to be in C. We can argue the same way that all d i have to be in C A by starting from the side of the path. This leads to a contradiction. d-separation implies t-separation: Suppose C d-separates A and. Partition C according to the following algorithm: 1. Define C 0 as follows: if there is a simple trek a u s u 0 v 0 v t c, such that a A, c C, u i, v j C, assign c to C 0. (a = u 0 is OK.) 2. Given C k, define Ck+1 as follows: if there is a simple trek c 1 u s u 0 v 0 v t c 2, such that c 1 C k, c 2 C\(C 0 Ck ), u i, v j C, assign c 2 to C k+1. Iterate this step until C k+1 =. 3. Let K be the last k such that C k. Let C = k=0,...,k C k and C A = C\C. The algorithm will end in a finite number of steps: in step 1 and each iteration of step 2 of the algorithm, only finitely many treks need to be considered and C is finite. Now we prove the partition (C A, C ) produced by the algorithm t-separates A C and C. Assume there is a trek u s u 0 v 0 v t (1) 4

5 between a vertex u s A C and a vertex v t C, such that no vertex on the A C side is in C A and no vertex on the C side is in C. That is u i C A, for i = 0,.., s and v j C, for j = 0,...,t. In particular u 0 v 0 C. We can assume the trek is simple, otherwise consider the loop-erased portion of the trek. Let i 0 be the smallest i such that u i A C and let j 0 be the smallest j such that v j C. I claim that v j0 C and v j0, which is a contradiction. If v j0 C, the sub-path of (1) between u i0 and v j0 satisfies the criterion in step 1 (if u i A) or 2 (if u i C ) of the partitioning algorithm, so v j0 should have been assigned to C, which contradicts that v j0 C. So v j0 has to be in. Next I show v j0 either. y construction, u i0 is in A or C. If u i0 A, the sub-path of (1) between u i0 and v j0 is a d-connecting path between a vertex in A and a vertex in. This is a contradiction. If u i0 C, there exists a sequence of treks as described in step 1 and 2 of the partitioning algorithm connecting a vertex in a A and u i0. Concatenating these treks with the sub-path of (1) between u i0 and v j0 gives a path between a and v j0 : a c 0 c r u i0 v j0 (2) where c i C for i = 0,...,r, a A, v j0, and all other vertices are not in C and are not colliders on the path. All colliders on the loop-erased part of (2) are either in C or have a descendant in C, so it is a d-connecting path between a and v j0 in. This is a contradiction. Acknowledgement I would like to thank Seth Sullivant and Thomas Richardson for helpful conversations and discussions. References Seth Sullivant and Kelli Talaska. Trek separation for Gaussian graphical models, URL Steffen L. Lauritzen. Graphical models. Oxford University Press, ISN