19. Poisson BIC Test

19.1. Summary

The Poisson BIC Test is simply the Poisson Prior 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 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 underlying likelihood score plus the Poisson Prior Score over structure. The difference in score is then thresholded at 0:

  • if the Poisson-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 Poisson-augmented BIC score prefers the graph with or without the edge.

19.2. When to use

  • You want a CI test that is tightly aligned with a Poisson-based structural prior on graphs.

  • You are already using PoissonPriorScore (or a BIC-like score plus a Poisson structural prior) for structure learning and want your CI tests to reflect the same sparsity assumptions.

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

19.3. Relation to Poisson Prior Score

The Poisson Prior Score is a structural prior over graph complexity. It places a Poisson distribution on edge or parent counts and adds the resulting log prior as a term in the score, on top of the usual likelihood or BIC-type term.

The Poisson BIC Test:

  • reuses that same score (likelihood + Poisson structural 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, Poisson BIC Test does not introduce a new noise model; it only reuses the existing Poisson-based structural prior as a test statistic.

19.4. Test details (conceptual)

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

  1. Constructs two models:

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

    • M_without: the corresponding model with that edge removed.

  2. Computes the Poisson-augmented score for each model:

    • score_with = base likelihood/BIC + PoissonPriorScore(M_with),

    • score_without = base likelihood/BIC + PoissonPriorScore(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 Poisson-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 rescaling to p-values; the decision is directly based on the sign of the score difference.

19.5. Parameters

The Poisson BIC Test exposes the same structural-prior parameters as PoissonPriorScore, restricted to what is needed for testing:

Parameter (camelCase)

Description

precomputeCovariances

Boolean. If true, precomputes and caches covariance (and possibly cross-covariance) matrices used by the underlying likelihood part of the score. This speeds up repeated evaluations at the cost of additional memory. If false, these quantities are recomputed on the fly, which saves memory but can be slower for large graphs or many score calls.

poissonLambda

Double > 0. Lambda parameter for the Poisson structural prior on the number of edges or parents. This controls the expected count under the prior. Smaller values favor sparser graphs (fewer edges or parents on average); larger values allow denser graphs. Default is 1.0; minimum is about 1e-10.

singularityLambda

Double. Handles singular or nearly singular covariance matrices in the underlying likelihood calculation. 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 0.0. Use a small positive value if you encounter numerical-singularity warnings.

19.6. Strengths

  • Fully aligned with the PoissonPriorScore: the CI decisions and score search share the same sparsity prior.

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

  • Naturally expresses a belief in Poisson-distributed complexity (for example, typical number of parents per node) directly in the CI testing step.

19.7. Limitations

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

  • The result depends on the choice of poissonLambda and the base score: overly small lambda can force over-sparse graphs (missing edges), while overly large lambda can allow too many edges.

  • Requires computing scores for two models per CI query, which can be more expensive than tests that only use local partial correlations or contingency tables.