Learning Applicability Envelopes for Conditional Independence Testing

Conditional independence (CI) testing is a foundational task in statistical inference, yet even for discrete data, classical CI-test methods fail in different and often incompatible regimes. This raises a practical question: *given a specific CI-testing instance, which test method should be trusted?* To study this question, we first synthesize verifiable data and post-train a large language model to combine the outputs of four representative CI-test methods, ${\chi }^2$, Fisher's exact, mutual information (MI), and logistic regression, into a final CI decision, based on the outputs of the CI tests and a four-dimensional feature vector characterizing the difficulty of the instance through observable data-quality and conditioning properties. We then analyze the router's learned policy in method-disagreement regimes and extract quantitative decision thresholds over the feature space to analyze **applicability envelopes** of the different CI-test methods. We validate these thresholds against reference transitions in test error rates and examine their variation across data regimes. Overall, our LLM router achieves the highest F1 on our synthetic evaluation set and shows competitive transfer performance with matched non-LLM routers on three additional Bayesian networks and the Sachs dataset. Furthermore, our analyses identify boundaries consistent with changes in test reliability and connect an MI boundary to the treatment of collapsed strata in the tested implementation's degrees of freedom. Finally, we evaluate the router on the Sachs flow-cytometry dataset at both the CI-testing level and the downstream causal-discovery level, where it achieves the best skeleton recovery among the evaluated PC variants.