38. PC-Max — PC with Maximum-p Collider Orientation

Type: Constraint-based
Output: CPDAG
Implements: Ramsey (2016)
Knowledge: Fully supported (same as PC)

PC-Max is an enhanced collider-orientation variant of the PC algorithm. It keeps the entire adjacency phase unchanged, but modifies the orientation of unshielded colliders by selecting the separating set with the maximum p-value among all valid separating sets. This greatly reduces false collider orientations and improves arrowhead precision—especially in high-dimensional or moderate-sample settings.

In Tetrad, PC-Max is the default configuration of PC.

38.1. Key Idea

PC-Max improves collider orientation while leaving all other PC operations unchanged.

  1. Collect all separating sets S such that X ⟂ Z | S.

  2. Compute p-values for each CI test.

  3. Choose the separating set with the maximum p-value.

  4. Orient X → Y ← Z iff Y ∉ S*.

All further orientations use standard Meek rules.

38.2. When to Use

  • You want higher orientation accuracy than standard PC.

  • Datasets are high-dimensional, noisy, or small-sample.

  • You want a drop-in replacement for PC with no extra settings.

38.3. Relation to Standard PC

  • Adjacency phase: identical

  • CI tests: identical

  • Orientation propagation: identical except collider selection

  • Knowledge support: identical

  • Output: CPDAG

See the PC page for details.

38.4. Prior Knowledge Support

PC-Max inherits full background knowledge support:

  • forbidden/required edges

  • temporal/tier constraints

  • partial orientations

All constraints are enforced consistently during adjacency removal and orientation.

38.5. Strengths

  • Markedly fewer false colliders

  • Higher arrowhead precision

  • No additional computation

  • No new parameters

  • Tetrad’s default and recommended method

38.6. Limitations

  • Same as PC: assumes DAGs without latent confounding

  • Faithfulness and suitable CI tests required

  • For latent confounders, use FCI/GFCI/BOSS-FCI/FCIT

38.7. Key Parameters in Tetrad

PC-Max adds no new parameters.
Enabled with:

colliderOrientationStyle = "maxP"

All other parameters match PC.

38.8. Reference

Ramsey, J. (2016).
Improving accuracy and scalability of the PC algorithm by maximizing p-value.
arXiv:1610.00378.

38.9. Summary

PC-Max = PC + maximum-p separating sets for collider identification.
A simple but powerful improvement that yields cleaner, more reliable CPDAG orientations.