15. m-Separation Test

15.1. Summary

The m-Separation Test is a graphical independence test that does not use data. Instead, it tests whether two variables X and Y are m-separated by a set S in a given graph (for example, a DAG, MAG, or PAG). It is used when the underlying independence information comes from a known causal graph rather than from statistical tests.

15.2. When to use

  • You have a known or assumed causal graph and want to compute implied independences X ⟂ Y | S directly from the graph structure.

  • You are debugging search algorithms or comparing learned graphs to ground truth.

  • You are using oracle experiments where m-separation plays the role of an independence oracle.

15.3. Assumptions

  • The graph is interpreted under standard rules of d-separation (for DAGs) or m-separation (for MAGs/PAGs).

  • The causal Markov and faithfulness assumptions relate graphical separation to statistical independence.

15.4. Test details (conceptual)

For each independence query X ⟂ Y | S, the test:

  1. Examines all paths between X and Y in the graph.

  2. Determines whether every such path is blocked given S by the rules of m-separation (colliders, non-colliders, descendant conditions, etc.).

  3. Returns “independent” if all paths are blocked, and “dependent” otherwise.

  4. No numeric statistic or p-value is computed; the output is exact given the graph.

15.5. Parameters in Tetrad

No Parameters.

15.6. Strengths

  • Exact given the graph; no sampling error.

  • Ideal for debugging algorithms and running simulation studies with a known ground truth.

  • Extremely fast for moderate graph sizes.

15.7. Limitations

  • Requires a known graph; not applicable when only data are available.

  • Assumes that independences correspond exactly to m-separation in the graph.

15.8. References

  • Spirtes, P., Glymour, C. N., & Scheines, R. (2000). Causation, Prediction, and Search (2nd ed.). MIT Press.

  • Zhang, J. (2008). On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias. Artificial Intelligence, 172(16–17), 1873–1896.