# Probabilistic Independence Test

## 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.

## 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.

## 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).

## 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.

## 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`. |

## Strengths

- Provides exact or near-exact independence decisions given the model.
- Ideal for algorithm evaluation and theoretical experiments.

## Limitations

- Not applicable when only raw data are available and no model is given.
- Results are only as good as the underlying model specification.