2. BOSS-FCI — Best-Order Score Search + FCI Refinement

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

BOSS-FCI is one of the mixed-strategy latent-variable algorithms introduced in
Ramsey, Andrews & Spirtes (2025). It combines:

1. BOSS — a score-based ordering search that produces a high-quality CPDAG, and

2. An FCI-style latent-variable correction stage, which transforms the CPDAG into a PAG.

The resulting algorithm is faster than GFCI and usually more accurate, particularly in medium- and high-dimensional settings.

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2.1. Key Idea

BOSS-FCI follows the general X-FCI template:

1. Score phase

   • Run BOSS to produce a CPDAG (causal sufficiency assumed for this step).

   • BOSS evaluates local SEM-BIC–type scores under different orderings, using memoization and efficient updates.

2. Latent-variable correction

   • Treat the BOSS CPDAG as an “oracle skeleton” and apply:

     • Orientation from separating sets

     • Possible-D-SEP pruning

     • Zhang-style orientation propagation (Zhang, 2008)

     • PAG legality checks (maximality, acyclicity, almost-cycles, endpoint corrections)

   • This yields a PAG that accounts for latent confounders and (optionally) selection bias.

   • BOSS-FCI is therefore a hybrid algorithm: it uses score-based reasoning for adjacency decisions and Zhang-style constraint-based logic for completing a PAG.

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2.2. When to Use

• You want a PAG but pure FCI is too noisy or too slow.

• BOSS performs well on your data (medium-to-large p, moderate sparsity).

• You want a drop-in alternative to GFCI with better scalability.

• You have mixed continuous/discrete data (BOSS supports mixed BIC scoring).

• You want a method that scales to dozens–hundreds of variables with latent confounding.

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2.3. Strengths

• More accurate than GFCI in most regimes (Ramsey et al. 2025).

• Fewer false positives and cleaner PAGs than standard FCI.

• Leverages BOSS’s robust adjacency selection.

• Parallelizable and efficient for larger variable counts.

• Fully knowledge-aware (forbidden edges, required edges, tiers, temporal constraints).

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2.4. Limitations

• BOSS phase assumes causal sufficiency, so the initial CPDAG can still suffer mis-orientations in heavily confounded models.

• Still requires CI tests for the correction phase (although fewer than GFCI).

• For extremely dense graphs, score-based phases can slow down.

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2.5. How It Differs From Related Algorithms

• vs. GFCI

  • Same conceptual pipeline but with BOSS replacing FGES.

  • Usually more accurate and efficient.

• vs. GRaSP-FCI

  • GRaSP uses permutation scoring; BOSS uses order-based BIC optimization.

  • BOSS-FCI is typically more stable on mixed/continuous data.

• vs. FCIT

  • FCIT uses targeted CI testing guided by BOSS and guarantees PAG legality,
    whereas BOSS-FCI still uses a traditional FCI refinement phase.

• **vs. LV-Heuristic **

  • LV-Heuristic heuristically converts a BOSS DAG directly to a PAG.

  • BOSS-FCI is a principled algorithm with sound PAG-correction steps.

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

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2.6. Prior Knowledge Support

BOSS-FCI fully supports knowledge.

You may connect a Knowledge box in the Tetrad GUI or provide a
Knowledge object programmatically.

Enforced constraints:

• Required edges

• Forbidden edges

• Tier / temporal ordering

• Forbidden ancestor/descendant relations

• Any additional Tetrad structural constraints

Knowledge is enforced during both:

• The BOSS adjacency/orientation decisions, and

• The FCI-style PAG refinement.

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2.7. Key Parameters in Tetrad

BOSS-FCI inherits parameters from BOSS and from the FCI refinement step.
CamelCase names (GUI / script API) shown.

2.7.1. BOSS-stage parameters

Parameter	Meaning
penaltyDiscount	BIC penalty multiplier (higher → sparser).
maxDegree	Maximum degree allowed during scoring.
numThreads	Parallelism level for scoring.
verbose	Print decision logs.

2.7.2. FCI-refinement parameters

Parameter	Meaning
depth	Max conditioning set size for CI tests.
stableFas	Order-independent adjacency removal.
excludeSelectionBias	Whether to disallow interpreting circles as selection bias.
verbose	Print CI tests / orientations.

(Exact list depends on the current Tetrad release.)

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2.8. 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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2.9. Summary

BOSS-FCI = BOSS CPDAG + FCI latent-variable correction.
A fast, accurate hybrid PAG learner that outperforms GFCI in most settings while remaining fully knowledge-aware and scalable.