  Conditional independence (CI) testing is a fundamental and challenging task
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; we refer to this class of tests as
regression-based tests. Although these methods are guaranteed to control Type-I
error when the supervised learning methods accurately estimate the regression
functions or Bayes predictors of interest, their behavior is less understood
when they fail due to misspecified inductive biases; in other words, when the
employed models are not flexible enough or when the training algorithm does not
induce the desired predictors. Then, we study the performance of
regression-based CI tests under misspecified inductive biases. 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 regression-based CI
test robust against misspecified inductive biases. Finally, we conduct
experiments with artificial and real data, showcasing the usefulness of our
theory and methods.
