27. GRaSP-FCI — Greedy Relaxations of Sparsest Permutation + FCI Refinement

Type: Hybrid (score-based permutation search + CI tests)
Output: PAG
Knowledge: Fully supported (all Tetrad knowledge constraints)

GRaSP-FCI is one of the mixed-strategy latent-variable algorithms introduced in
Ramsey, Andrews & Spirtes (2025). It follows the same X-FCI template as BOSS-FCI and GFCI, but uses GRaSP—a permutation-based score search—for the initial CPDAG before performing an FCI-style latent-variable correction.

The result is a PAG that often shows better adjacency recall than FGES-based hybrids and more stable performance on certain permutation-favorable models.

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27.1. Key Idea

GRaSP-FCI has two stages:

1. Score-based permutation search (GRaSP)

   • GRaSP searches over variable permutations and evaluates their implied DAGs using a sparsity-oriented SEM-BIC score.

   • Rather than exploring graph space directly, it uses the fact that every permutation induces a unique DAG skeleton.

   • The best-scoring permutation yields a CPDAG to serve as the skeleton for refinement.

2. Latent-variable correction (FCI refinement)

   • Starting from the GRaSP CPDAG, apply the full set of FCI refinement procedures:

     • Possible-D-SEP checks

     • Sepset-based collider identification

     • Orientation propagation rules

     • PAG legality restoration

   • Produces the final PAG.

GRaSP-FCI thus combines permutation scoring (which behaves well for some data regimes) with robust FCI latent reasoning.

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27.2. When to Use

• You expect the model to benefit from permutation-based scoring rather than purely order-based scoring.

• GRaSP performs well empirically on your dataset (especially moderate-sized continuous data).

• You want a PAG but find GFCI too noisy or FGES-derived skeletons too brittle.

• You want a hybrid method that preserves causal discovery under latent confounding while leveraging permutation sparsity.

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27.3. Strengths

• Often superior adjacency discovery on datasets where permutation DAGs capture sparsity better than order-based DAGs.

• More stable than FCI alone; fewer spurious adjacencies.

• Fully knowledge-aware.

• Leverages GRaSP’s nonparametric flexibility and sparsity logic.

• PAG refinement ensures latent confounders and selection bias are handled.

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27.4. Limitations

• Permutation scoring can be slower than order-based BOSS for large p.

• Still requires CI tests in the refinement stage.

• Accuracy depends strongly on permutation quality in the GRaSP step.

• Does not guarantee legality during scoring — legality is restored only in the FCI stage.

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27.5. How It Differs From Related Algorithms

• vs. BOSS-FCI

  • GRaSP-FCI uses permutations, not variable orders.

  • Typically slower than BOSS-FCI; sometimes better on medium-sized datasets with specific structure.

• vs. GFCI

  • GFCI uses FGES for skeleton discovery; GRaSP-FCI uses GRaSP.

  • GRaSP-FCI can outperform when score-based permutation sparsity is informative.

• vs. FCIT

  • FCIT directs CI tests using score priorities; GRaSP-FCI still uses a classical FCI refinement.

  • FCIT guarantees PAG legality at all steps; GRaSP-FCI guarantees legality only at the end.

• vs. LV-Heuristic

  • LV-Heuristic is a heuristic PAG-from-DAG transformer.

  • GRaSP-FCI is a principled hybrid algorithm with full correctness of FCI refinement.

Cross-references:
👉 GRaSP •
👉 BOSS-FCI •
👉 GFCI •
👉 FCIT •
👉 LV-Heuristic

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27.6. Prior Knowledge Support

GRaSP-FCI is fully knowledge-aware.

You may supply a Knowledge object in the GUI or programmatically. Constraints apply to:

• Permutation scoring (forbidden ancestors/descendants, tiers)

• CPDAG construction

• FCI refinement (required/forbidden edges, temporal structure)

Knowledge is respected throughout all phases.

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27.7. Key Parameters in Tetrad

GRaSP-FCI combines parameters from:

27.7.1. GRaSP scoring stage

Parameter	Meaning
penaltyDiscount	BIC penalty multiplier (sparsity control).
maxDegree	Restricts search to graphs with limited degree.
numThreads	Parallel evaluation of permutations.
verbose	Print detailed scoring decisions.

27.7.2. FCI refinement stage

Parameter	Meaning
depth	Maximum conditioning-set size for CI tests.
stableFas	Order-invariant adjacency removal.
excludeSelectionBias	Whether to interpret circle–circle as selection structures.
verbose	Print refinement steps and orientations.

(Exact parameter list may vary by Tetrad version.)

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27.8. Reference

Ramsey, J., Andrews, B., & Spirtes, P. (2025).
Efficient Latent Variable Causal Discovery: Combining Score Search and Targeted Testing.
arXiv:2510.04263.

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27.9. Summary

GRaSP-FCI = GRaSP permutation CPDAG + FCI latent-variable refinement.
A hybrid PAG learner that combines permutation sparsity with principled FCI logic, producing cleaner PAGs than FCI and often more reliable adjacencies than GFCI.