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L4-02 · DAGs and d-separation

Use path rules to decide which conditional independences a causal graph implies.

Level 4 icon Level 4 · Undergraduate
Objective. Use path rules to decide which conditional independences a causal graph implies.
Before this lesson. L4-01, L2-07
Hands-on lab. Confounding and intervention

Learn

A directed acyclic graph, or DAG, contains directed edges and no directed cycle. Each node names a variable. An edge points from a direct cause to its child in the modeled system. A missing edge is a modeling claim, rather than proof that no relationship exists.

A path ignores edge direction while tracing connections. A noncollider on a path blocks it when conditioned on. A collider has two arrowheads meeting at that node. It blocks the path unless the collider or one of its descendants is conditioned on. Apply the rules to every path.

Two node sets are d-separated by a conditioning set when every connecting path is blocked. Under the causal Markov assumption, this graphical relation implies a conditional independence in the distribution. The reverse implication requires more assumptions, such as faithfulness. Hidden causes can require a graph with additional nodes.

Practice

Solution

Without conditioning, the fork path is open, so A and B are not d-separated. Conditioning on the noncollider Z blocks the only path and d-separates them.

Check your understanding

What does d-separation guarantee under the Markov assumption?

Show the answer

A. A conditional independence

D-separation implies the corresponding conditional independence under the Markov assumption. It does not establish a nonzero effect or identify a unique graph.

Sources

This lesson draws on Pearl, 2009Spirtes et al., n.d..

Next: L4-03 · Confounders, mediators, and colliders

References
  1. Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3(none), 96–146. 10.1214/09-ss057
  2. Spirtes, Glymour, & Scheines. (n.d.). Causation, Prediction, and Search. https://www.cs.cmu.edu/afs/cs.cmu.edu/project/learn-43/lib/photoz/.g/web/.g/scottd/fullbook.pdf
  3. Bareinboim, E. (2025). Causal Artificial Intelligence: A Roadmap for Building Causally Intelligent Systems. https://causalai-book.net/