39. PCD — PC for Deterministic Relations

Type: Constraint-based
Output: CPDAG
Reference:
Glymour, C. (2007). Learning the structure of deterministic systems.
In Causal Learning: Psychology, Philosophy, and Computation, pp. 231–240.

Pcd is a variant of the PC algorithm designed to handle deterministic or near-deterministic functional relationships—situations where one variable is an exact or almost exact function of others (e.g., sums, ratios, duplicates, encoded transformations). Such relations violate the faithfulness assumption underlying PC and can lead to spurious edges.
Pcd adds safeguards to prevent deterministic relations from corrupting adjacency search and collider orientation.

39.1. Key Idea

Determinism breaks the usual CI patterns:

If (Y = f(X)), then (X) and (Y) are never independent—even when conditioning would ordinarily separate them under faithfulness.

To address this, Pcd introduces:

  1. Detection of (near-)deterministic relations
    Identifies variables involved in deterministic mappings using heuristic or test-based diagnostics.

  2. Suppression of invalid CI tests
    Avoids CI tests that would be misleading under determinism (e.g., tests that should show independence but cannot).

  3. Adjusted adjacency pruning
    Prevents deterministic variables from forcing large, spurious cliques in the skeleton.

  4. PC-style orientation (CPC-consistent)
    Uses standard PC/CPC orientation rules but applied to the cleaned skeleton.

The result is a CPDAG that more faithfully reflects causal relations even when determinism is present.

39.2. When to Use

Use Pcd when you suspect or know that:

  • Variables include exact deterministic functions, such as:

    • sums, differences, averages

    • ratios or percentages

    • logical or categorical encodings

  • There is near-perfect multicollinearity (correlations ≈ ±1).

  • Standard PC produces large cliques or spurious adjacencies around deterministic chains.

  • You want a PC-like method that is robust to faithfulness violations induced by determinism.

39.3. Prior Knowledge Support

Pcd fully supports Tetrad background knowledge:

  • Required edges

  • Forbidden edges

  • Tier/temporal constraints

All knowledge is respected during both adjacency search and orientation.

39.4. Strengths

  • Robust to determinism
    Handles exact or near-exact functional dependencies.

  • Cleaner skeletons
    Removes artificial cliques introduced by deterministic relations.

  • PC-compatible
    Same interface, same parameters, and same interpretation as PC/CPC.

  • Works with all CI tests
    Fisher Z, G-test, KCI/RCIT, basis-function tests, etc.

39.5. Limitations

  • Does not model latent confounders
    (use FCI, RFCI, GFCI, FCIT, etc. for that)

  • Determinism must be detectable
    Very subtle determinism can still cause issues.

  • Still a constraint-based method
    Thus inherits finite-sample CI-test sensitivity from PC/CPC.

39.6. Key Parameters in Tetrad

Parameter (camelCase)

Description

stableFas

Use order-independent adjacency search.

colliderOrientationStyle

PC, PC-Max, or CPC-style collider detection.

depth

Maximum conditioning-set size.

fdrQ

Optional FDR control (replaces α).

verbose

Print detailed CI-test and orientation logs.

39.7. Summary

Pcd is a determinism-aware variant of PC that prevents faithfulness violations from creating spurious edges. It yields cleaner, more interpretable CPDAGs when datasets include exact or near-exact functional relationships, while retaining the familiar behavior of PC/CPC.