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.

在现代统计学和机器学习中，条件独立性检验是基础性且具有挑战性的。许多现代的条件独立性检验方法依赖于强大的监督学习方法，在学习回归函数或贝叶斯预测器时作为一种中间步骤。然而，当监督学习方法由于模型错误估计导致失败时，这些方法的行为了解还很有限。在更广义上，即使使用通用逼近器（如深度神经网络），模型错误估计仍然可能产生。因此，我们研究了基于回归的条件独立性测试在模型错误估计下的性能。具体地，我们提出了三个基于回归的测试的测试误差的新近似值或上界，这些误差依赖于模型错误估计。此外，我们引入了一种新的基于回归的条件独立性测试方法，即 Rao-Blackwellized 预测器测试（RBPT），该方法对模型错误估计具有鲁棒性。最后，我们使用人工数据和真实数据进行实验证明了我们的理论和方法的有用性。