20. FGES-MB — FGES Markov Blanket Search

Type: Score-based, local / Markov blanket
Output: CPDAG (around a target)

FgesMb is a local variant of FGES that focuses on the Markov blanket of a single target variable rather than the full graph.
Given a target T, it runs a greedy equivalence search but concentrates scoring and edge updates on the neighborhood of T, returning a local CPDAG that encodes all candidate Markov blankets of T consistent with the score.

It is the score-based counterpart of PcMb: instead of conditional independence tests and a significance level, FgesMb uses BIC-type scores to select and orient edges.

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20.1. Key idea

Use FGES-style greedy search, but restrict attention to edges that matter for the target T:

• In a DAG, the Markov blanket of T consists of:

  • the parents of T,

  • the children of T,

  • and the parents of those children.

• Instead of learning a full CPDAG over all variables, FgesMb:

  • runs a score-based forward–backward search,

  • but prioritizes changes that affect the local structure around T,

  • and returns a CPDAG whose neighborhood around T encodes all Markov blankets compatible with the score.

Internally, the search uses the same ideas as FGES:

1. Forward phase: greedily add edges that give the largest score improvement, subject to acyclicity and background knowledge.

2. Backward phase: greedily remove edges that most improve the score.

3. Equivalence-class representation: maintain and update a CPDAG rather than a single DAG.

FgesMb adopts these mechanisms but is tuned for local Markov blanket recovery instead of global structure.

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20.2. When to use FgesMb

Use FgesMb when:

• You care primarily about one target variable T (for example, an outcome or label).

• You prefer a score-based approach (BIC, mixed BIC, etc.) rather than CI tests.

• The number of variables is large and learning a full CPDAG would be expensive or unnecessary.

• You want a principled, score-based Markov blanket for downstream tasks like regression or classification.

Typical applications:

• Feature selection for supervised learning, where the goal is to identify a causally motivated feature set around a target.

• High-dimensional problems where global structure learning (full FGES, BOSS, GRaSP) is too expensive.

• Comparative studies: PcMb vs FgesMb, CI-based vs score-based Markov blankets.

If you need global structure, you would normally use FGES, BOSS, or GRaSP instead.

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20.3. Prior knowledge support

Does it accept background knowledge?
Yes. FgesMb respects the same knowledge constraints as FGES:

• Required edges

  • Force certain arrows to be present (for example, “X must cause T”).

• Forbidden edges

  • Disallow particular adjacencies or orientations.

• Tiers / temporal constraints

  • Enforce a partial order over variables, so that edges must go from earlier to later tiers.

All search operations (adds/removals) are restricted to be consistent with this knowledge.

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20.4. Strengths

• Local and target-focused

  • Efficient when you only care about one target T, not the entire graph.

• Score-based semantics

  • Uses BIC-type scores instead of CI tests, which can be appealing when:

    • sample sizes are large,

    • model assumptions (for example, linear Gaussian, discrete multinomial) are reasonable.

• Causal Markov blanket interpretation

  • The local CPDAG around T encodes all DAGs (and hence all Markov blankets) consistent with the score and knowledge.

• Comparable to FGES

  • Inherits FGES optimizations (caching, heuristic speedups, parallelism), making it usable in moderately high dimensions.

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20.5. Limitations

• Model assumptions

  • The score typically assumes:

    • linear Gaussian SEMs for continuous data,

    • multinomial models for discrete data,

    • or a specified mixed-data model.

  • Misspecification (for example, strong nonlinearities) can degrade performance.

• Heuristic nature

  • Greedy search may get stuck in local optima.

  • Heuristic speedups (for example, correlation pre-screening) can trade off exactness for speed.

• Finite-sample sensitivity

  • As with any score-based method, sampling noise can:

    • add spurious neighbors,

    • or miss true neighbors of T.

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20.6. Key parameters in Tetrad

Exact names can vary slightly between GUI and code, but conceptually FgesMb exposes the same controls as FGES plus a target variable.

Parameter (camelCase)	Description
targetName	The distinguished target variable whose Markov blanket is to be learned.
scoreType	Choice of score (for example, SemBicScore, DiscreteBicScore, MixedBicScore).
penaltyDiscount	Multiplier on the complexity penalty (larger values favor sparser blankets).
maxDegree	Optional cap on the maximum number of neighbors any node (including T) may have.
useHeuristicSpeedup	Enable correlation-based edge pre-screening.
numThreads	Number of threads for parallel scoring.
verbose	Print progress and score changes.

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20.7. Reference

FgesMb is a local Markov blanket variant of the FGES algorithm:

• Ramsey, J., Glymour, M., Sanchez-Romero, R., & Glymour, C. (2017).
  A million variables and more: the fast greedy equivalence search algorithm for learning high-dimensional graphical causal models, with an application to functional magnetic resonance images.
  International Journal of Data Science and Analytics, 3, 121–129.

This paper introduces and studies FGES; FgesMb applies the same score-based ideas to Markov blanket discovery for a single target.

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20.8. Summary

FgesMb is a score-based Markov blanket learner built on FGES: it focuses on a single target T, runs a greedy equivalence search tuned to the local neighborhood of T, and returns a CPDAG encoding all score-consistent Markov blankets of T under your chosen BIC-type score and knowledge constraints.