7. Conservative PC (CPC) — Conservative Collider Orientation

Type: Constraint-based (conservative variant)
Output: e-pattern (equivalence class of CPDAGs)

Conservative PC (CPC) is a modification of the PC algorithm designed to avoid false-positive collider orientations by requiring stronger evidence before declaring a triple a collider. Instead of orienting all unshielded colliders using a single separating set, CPC checks all relevant separating sets and only orients a collider when all such tests agree.

This prevents erroneous collider orientations when CI tests are noisy or unstable.

7.1. Key Idea

Given an unshielded triple:

X — Y — Z     with    X not adjacent to Z

PC orients this as a collider X → Y ← Z if Y ∉ sepset(X, Z).

CPC strengthens this:

7.1.1. Conservative Collider Rule

For every separating set S such that
X ⟂ Z | S and S is a subset of adj(X) ∪ adj(Z):

  • If Y ∉ S for all such S, conclude:
    X → Y ← Z (collider)

  • If Y ∈ S for all such S, conclude:
    X — Y — Z (noncollider)

  • Otherwise:
    Leave the triple unoriented (ambiguous)

The resulting graph is an e-pattern rather than a CPDAG—some orientations remain intentionally unresolved.

7.2. When to Use

Use CPC when:

  • You expect small sample sizes, noisy CI tests, or unstable separating sets

  • False-positive collider orientations would be costly (e.g., downstream adjustment sets)

  • You want a more conservative, robust version of PC

CPC is strictly more conservative than PC-Max and classic PC.

7.3. Prior Knowledge Support

CPC fully supports background knowledge, including:

  • Required edges

  • Forbidden edges

  • Tier/temporal constraints

  • Any other Knowledge box logic in Tetrad

All constraints are respected during adjacency and orientation phases.

7.4. Strengths

  • Greatly reduces false-positive collider orientations

  • Robust under sampling variability

  • Produces a safe e-pattern that captures uncertainty

  • Still quite fast in practice

7.5. Limitations

  • May leave many triples unoriented (intentionally)

  • Produces an e-pattern rather than a CPDAG, so fewer directions may be implied

  • Conservative nature may propagate to downstream inference

7.6. Key Parameters in Tetrad

CPC is implemented as a collider orientation style inside the PC search wrapper, so it shares PC’s parameters:

Parameter (camelCase)

Description

stableFas

Use stable (order-independent) adjacency search.

colliderOrientationStyle

Set to “Conservative” to use CPC rules.

allowBidirected

Allow temporary bidirected edges.

depth

Maximum conditioning set size.

fdrQ

FDR-controlled CI testing option.

timeLag

Lag structure for time-series data.

timeLagReplicatingGraph

Replicate lag structure across slices.

verbose

Log CI tests and orientation steps.

See the main PC documentation for general parameter behavior.

7.7. Reference

The original peer-reviewed publication:

Ramsey, J., Zhang, J., & Spirtes, P. (2006).
Adjacency-faithfulness and conservative causal inference.
In Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence (UAI-06), pp. 401–408.

A later archival version:
Ramsey, J., Zhang, J., & Spirtes, P. L. (2012). Adjacency-faithfulness and conservative causal inference. arXiv:1206.6843.

7.8. Summary

Conservative PC provides a safe, cautious variant of PC, orienting only those colliders supported unanimously by all relevant separating sets. It is ideal when avoiding false orientations is more important than maximizing orientation coverage.