When a causal graph contains an open path between two variables after conditioning on Z, the variables are d-connected. The graph therefore indicates potential dependence in its modeled joint distribution, without claiming that one variable causes the other. section 4.2 section 4.2.1
A path ignores arrow orientation when it is enumerated: chains, forks, and colliders all count as paths. D-connection then depends on how each path is blocked by the conditioning set. section 4.2.1
Check each path using these rules:
- A non-collider in a chain or fork blocks the path when it belongs to Z. section 4.2.1
- A collider such as i → m ← j blocks the path by default when neither it nor a descendant belongs to Z. section 4.2.1
- Conditioning on a collider or its descendant can open the path and create d-connection. section 4.2.1
- Two nodes are d-connected when at least one path remains open; they are d-separated only when every path is blocked. section 4.2.1
For causal analysis, enumerate paths between a treatment and outcome, then identify conditioning variables that block non-causal routes without accidentally opening collider paths. Graph surgery removes incoming edges to an intervention target, leaving paths that represent the target’s causal influence. section 4.1.2 section 7.2.4
Minimal separating sets
Suppose U and V have direct paths U → I → V and U → J → V, plus U → J → I → V. Conditioning on {I, J} blocks the two shorter paths and consequently blocks the longer path. In a graph with U ← I → V, U ← M → V, and U ← K → V, {I, M, K} is required for those paths; remaining routes may require J or L. Adding a collider to block one route can open another, so every affected path must be checked again. section 4.2.2
