21. Probabilistic Independence Test

21.1. Summary

The Probabilistic Independence Test is a wrapper test that uses explicit probability or density models (for example, from an instantiated Bayesian network or parametric model) to answer independence queries X ⟂ Y | S. It is used when the probability model, not the data, is considered the source of truth.

21.2. When to use

• You have a fully specified probabilistic model (e.g., a Bayes net or parametric SEM) and want to query its implied independences.

• You are performing oracle-style experiments where the model encodes the ground truth.

• You wish to test search algorithms in controlled synthetic settings.

21.3. Assumptions

• The provided probabilistic model is correct (for the purpose of the experiment).

• Independence decisions are made by checking whether the model assigns the same conditional distribution to X given S, with and without conditioning on Y (or equivalent factorizations).

21.4. Test details (conceptual)

For each X ⟂ Y | S query, the test:

1. Uses the underlying probability model to compute or compare P(X | S) and P(X | Y, S), or an equivalent characterization.

2. Declares independence if these distributions are equal (within numerical tolerance), and dependence otherwise.

3. Does not rely on raw data; the model itself is the oracle.

21.5. Parameters

Parameter (camelCase)	Description
noRandomlyDeterminedIndependence	Boolean. If true, independence decisions are made deterministically using the cutoffIndTest threshold: if the estimated probability of independence is above the cutoff, the variables are treated as independent; otherwise they are treated as dependent. If false (default), the test is allowed to use randomized decisions rather than a strict hard cutoff. Default is false.
cutoffIndTest	Double in [0.0, 1.0]. Independence cutoff threshold. When noRandomlyDeterminedIndependence = true, independence is declared whenever the estimated probability (or score) of independence exceeds this value. Default is 0.5.
priorEquivalentSampleSize	Double ≥ 1.0. Prior equivalent sample size for the underlying Bayesian model used to estimate independence probabilities. This acts like a pseudo-count total that is distributed across cells in the relevant contingency or parameter tables. Larger values make the prior stronger relative to the data (smoother probabilities); smaller values let the data dominate more. Default is 10.0.

21.6. Strengths

• Provides exact or near-exact independence decisions given the model.

• Ideal for algorithm evaluation and theoretical experiments.

21.7. Limitations

• Not applicable when only raw data are available and no model is given.

• Results are only as good as the underlying model specification.