Conditional independence (CI) testing is fundamental and challenging in modern statistics and machine learning. Many modern methods for CI testing rely on powerful supervised learning methods to learn regression functions or Bayes predictors as an intermediate step. Although the methods are guaranteed to control Type-I error when the supervised learning methods accurately estimate the regression functions or Bayes predictors, their behavior is less understood when they fail due to model misspecification. In a broader sense, model misspecification can arise even when universal approximators (e.g., deep neural nets) are employed. Then, we study the performance of regression-based CI tests under model misspecification. Namely, we propose new approximations or upper bounds for the testing errors of three regression-based tests that depend on misspecification errors. Moreover, we introduce the Rao-Blackwellized Predictor Test (RBPT), a novel regression-based CI test robust against model misspecification. Finally, we conduct experiments with artificial and real data, showcasing the usefulness of our theory and methods.

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