Conditional independence testing under misspecified inductive biases

TL;DR

This paper investigates the robustness of regression-based conditional independence tests when models are misspecified, proposing new bounds and a robust test to improve reliability in practical scenarios.

Contribution

It introduces the Rao-Blackwellized Predictor Test (RBPT) and provides theoretical bounds for testing errors under model misspecification, enhancing CI testing reliability.

Findings

RBPT is robust against misspecified models.

New bounds improve understanding of test errors under bias.

Experiments validate theoretical insights with real and artificial data.

Abstract

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.