Learning Kernelized Hypothesis for Hidden Confounder Detection

Abstract

Detecting hidden confounding is crucial for reliable causal analysis from observational data, directly determining which downstream causal inference method to be deployed. Inspired by the theory of higher-order regression, recent sample-efficient hypothesis testing strategies overcome the restrictive requirement of multiple heterogeneous data environment. Despite their progress on single-environment confounder detection, such methods suffer from intrinsic flaws that the structural functions of the causal models should be specified in prior (linear or specific kernel functions). By contrast, real-world data acquisition exhibits diverse, unknown forms of structural functions, imposing an important but challenging gap between theories of higher-order regressions to practical confounder detection. In this paper, we contribute a Bi-level Kernel Confounder Detection (BiKCD) framework by learning adaptive kernelized structural space of structural functions. Subsequently, our BiKCD constructs hypothesis testing by comparing coefficients from the higher-order regression and the classical ordinary least squares in learned kernelized space. Finally, the hypothesis is calibrated to ensure valid inference under adaptivity. Theoretically, we establish an oracle-type risk bound for the selected structural space over a candidate kernel family, with the Type-I error control for the downstream test. Extensive experiments on synthetic and real-world datasets demonstrate the effectiveness of the proposed BiKCD.