24. SEM BIC Test

24.1. Summary

The SEM BIC Test is simply the SEM BIC Score used as a conditional-independence decision rule.

For each query of the form “X independent of Y given S”, the test compares two models (or local graphs):

  • a model where the edge between X and Y is present (given S), and

  • a model where that edge is absent.

Both models are evaluated using the same SEM BIC Score (linear Gaussian SEM likelihood plus BIC-type penalty and any chosen structure prior).
The difference in score is then thresholded at 0:

  • if the SEM-BIC-based score difference is less than 0, the test treats
    X and Y as independent given S;

  • otherwise, it treats them as dependent.

There is no explicit alpha parameter; the decision is purely based on whether the SEM BIC score prefers the model with or without the edge.

24.2. When to use

  • You want a CI test that is tightly aligned with the SEM BIC Score used by your search algorithm (FGES, BOSS, GRaSP, etc.).

  • You are already using Sem BIC Score for structure learning and want your CI tests to reflect the same linear-Gaussian modeling assumptions and penalty.

  • You prefer a score-difference rule instead of p-values.

24.3. Relation to SEM BIC Score

The Sem BIC Score is a standard linear-Gaussian SEM BIC criterion, possibly augmented with a structure prior and different penalty rules. The SEM BIC Test:

  • reuses that same score (likelihood + BIC penalty + optional structure prior),

  • evaluates a model with the edge and a model without the edge,

  • then decides independence by checking whether the score difference is negative (BIC < 0 threshold).

So, SEM BIC Test does not introduce a new model or extra parameters; it just turns the Sem BIC Score into a CI test with a fixed zero score-difference threshold.

24.4. Test details (conceptual)

For each X, Y, and conditioning set S, the test:

  1. Constructs two models:

    • M_with: a model (or local SEM) including the edge between X and Y conditioned on S.

    • M_without: the corresponding model with that edge removed.

  2. Computes the SEM BIC Score for each model:

    • score_with = Sem BIC Score(M_with),

    • score_without = Sem BIC Score(M_without).

  3. Forms the score difference (for example, score_with − score_without).

  4. If this score difference is less than 0, then the edge is not supported by the SEM-BIC criterion, and the test declares:

    X and Y are independent given S.

  5. Otherwise, the test declares:

    X and Y are dependent given S.

There is no mapping to p-values; the decision is directly based on the sign of the BIC score difference.

24.4.1. Parameters

Parameter (camelCase)

Description

penaltyDiscount

Double ≥ 0.0. Penalty multiplier “c” in the BIC-type score used inside the test (for example, a score of the form 2·log-likelihood − c·k·log(N), where k is the number of free parameters and N is the sample size). Larger values impose a stronger complexity penalty and tend to favor sparser graphs; smaller values allow denser graphs.

structurePrior

Double ≥ 0.0. Structure prior coefficient controlling a binomial-style prior on the number of parents per node. When 0.0, the test uses essentially a flat structure prior. Positive values encode a preference for certain in-degree patterns (typically sparser graphs).

precomputeCovariances

Boolean. If true, precomputes and caches covariance (and possibly cross-covariance) matrices used by the internal Sem BIC Score. This speeds up repeated scoring at the cost of additional memory. If false, covariances are computed on the fly, which saves memory but may be slower for large graphs or many CI queries.

singularityLambda

Double. Handles singular or nearly singular covariance matrices. If singularityLambda > 0, that value is added to the diagonal (a ridge term) to stabilize matrix inverses. If singularityLambda < 0, a pseudoinverse is used instead. Default is often 0.0; use a small positive value if you encounter numerical-singularity warnings.

24.5. Strengths

  • Perfectly aligned with SEM BIC: CI decisions and score search share the same modeling assumptions and penalty.

  • No need to choose an alpha level; the decision rule is fixed at a zero score-difference threshold.

  • Exploits a familiar, well-studied criterion (BIC) for CI testing in linear Gaussian SEMs.

24.6. Limitations

  • There is no p-value calibration; decisions are based solely on score differences and the chosen penalty and structure prior.

  • The result depends on the choice of penaltyDiscount, semBicStructurePrior, and semBicRule. Very aggressive penalties can force over-sparse graphs (missing edges), while weak penalties can allow too many edges.

  • Requires computing scores for two models per CI query, which can be more expensive than tests such as Fisher Z that only use local partial correlations.