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Across the BoundariesExtrapolation in Biology and Social Science$

Daniel Steel

Print publication date: 2007

Print ISBN-13: 9780195331448

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780195331448.001.0001

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C. D‐SEPARATION

C. D‐SEPARATION

Source:
Across the Boundaries
Publisher:
Oxford University Press

D‐separation is a graphical concept whose interest lies in the following, highly nontrivial fact: for directed acyclic graphs, d‐separation indicates exactly those conditional and marginal probabilistic independencies entailed by the causal Markov condition (CMC) (cf. Pearl 2000, 18).4 A graph consists of a set of nodes, some or all of which are linked by lines, or edges. Typically, the nodes are understood to represent variables. A graph is said to be directed if each edge has an arrowhead at exactly one end. For example, consider the graphs in Figure A.1.

Graph (A) is directed, but (B) and (C) are not: (B), because of the undirected edge between W and N, and (C), because of the double‐headed arrow between W and S. A graph is said to be acyclic if it does not contain any sequence of arrows all aligned head to tail that begin and end at the same node. For example, the graph in Figure A.2 contains a cycle.

In contrast, graph (A) in Figure A.1 is both directed and acyclic.

D‐separation, then, is defined as follows:

  • A path p is said to be d‐separated (or blocked) by a set of nodes S if and only if

    1. p contains a chain i → m → j or a fork i ← m → j such that the middle node m is in S, or

  • 2. p contains an inverted fork (or collider) i → m ← j such that the middle node m is not in S and such that no descendant of m is in S.

    A set S is said to d‐separate X from Y if and only if S blocks every path from a node in X to a node in Y. (Pearl 2000, 16–17).

    X and Y are said to be d‐connected by a set S if S does not d‐separate them.

Notice that item 1 of the definition corresponds to the screening‐off rule (see sections 4.4.1 and 9.1). Meanwhile, item 2 corresponds to the rule that

C. D‐SEPARATION

Figure A.1 Directed versus undirected graphs

(p.209)
C. D‐SEPARATION

Figure A.2 A directed graph with a cycle

C. D‐SEPARATION

Figure A.3 An illustration of d‐separation

conditioning on colliders (or their descendants) may induce probabilistic dependence (see Figure 9.2 in section 9.1).

As an illustration, consider the directed acyclic graph in Figure A.3.

In this graph there are four paths between V and Z:

  1. (1) VWYZ

  2. (2) VWXZ

  3. (3) VXWYZ

  4. (4) VXZ.

Suppose that S is the empty set. Then paths (1), (3), and (4) are not blocked by S, though (2) is owing to the collider at X. Therefore, V and Z are d‐connected by the empty set. Recall that the faithfulness condition (FC) asserts that the only probabilistic independence relations are those entailed by the CMC. Thus, if the graph satisfies the FC, V and Z are probabilistically dependent when no variables are conditioned upon (i.e., they are marginally dependent).

Suppose that S = {X, Y}. Then paths (1), (3), and (4) are blocked by S. However, path (2) is not, since X is a collider on that path and X is S. Hence, V and Z are d‐connected by {X, Y}. Thus, if the graph satisfies the FC, then V and Z are probabilistically dependent conditional on this set of variables.

Finally, suppose that S = {W, X, Y}. In this case, all four paths are blocked by S. Thus, {W, X, Y} d‐separates V from Z. Consequently, if the graph in Figure A.3 satisfies the CMC, then V and Z are probabilistically independent conditional on {W, X, Y}.

D‐separation also characterizes exactly those probabilistic independencies entailed by linear cyclic structures with independent error terms (Richardson and Spirtes [1999]). That result is of particular interest because d‐separation and the CMC do not coincide for cyclic directed graphs. To see how this is so, consider the graph in Figure A.4.

(p.210)

C. D‐SEPARATION

Figure A.4 How d‐separation and the CMC differ for cyclic graphs

Note that Z is not a descendant of X, and the only parent of X is Y. Hence, if this graph satisfied the CMC, then X would be probabilistically independent of Z conditional on Y. However, {Y} does not d‐separate X from Y, since Y is a collider on the path X → Y ← Z. Given the proof that d‐separation characterizes exactly those independence relationships entailed by linear cyclic structures with independent error terms, the natural conclusion is that d‐separation is a more trustworthy guide for cyclic graphs than the CMC.

Notes:

(4.)  The CMC was discussed in sections 4.4.1 and 9.1.