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Detecting Unobserved Confounders: A Kernelized Regression Approach

Yikai Chen, Yunxin Mao, Chunyuan Zheng, Hao Zou, Shanzhi Gu, Shixuan Liu, Yang Shi, Wenjing Yang, Kun Kuang, Haotian Wang

TL;DR

KRCD tackles unobserved confounding in nonlinear single-environment observational data by embedding $T$, $X$, and $Y$ into an RKHS and comparing two kernelized regressions: standard KLS and HKLS. The method establishes that population regression coefficients coincide under $H_0$ (no confounding) and that finite-sample differences follow a zero-mean Gaussian distribution, enabling a statistically sound test. Empirically, KRCD outperforms baselines in NSE and competitive baselines in multi-environment settings, while delivering superior computational efficiency and robustness to kernel choices. This approach broadens confounding detection to nonlinear regimes without requiring instruments or multiple environments, with practical implications for causal inference in observational studies.

Abstract

Detecting unobserved confounders is crucial for reliable causal inference in observational studies. Existing methods require either linearity assumptions or multiple heterogeneous environments, limiting applicability to nonlinear single-environment settings. To bridge this gap, we propose Kernel Regression Confounder Detection (KRCD), a novel method for detecting unobserved confounding in nonlinear observational data under single-environment conditions. KRCD leverages reproducing kernel Hilbert spaces to model complex dependencies. By comparing standard and higherorder kernel regressions, we derive a test statistic whose significant deviation from zero indicates unobserved confounding. Theoretically, we prove two key results: First, in infinite samples, regression coefficients coincide if and only if no unobserved confounders exist. Second, finite-sample differences converge to zero-mean Gaussian distributions with tractable variance. Extensive experiments on synthetic benchmarks and the Twins dataset demonstrate that KRCD not only outperforms existing baselines but also achieves superior computational efficiency.

Detecting Unobserved Confounders: A Kernelized Regression Approach

TL;DR

KRCD tackles unobserved confounding in nonlinear single-environment observational data by embedding , , and into an RKHS and comparing two kernelized regressions: standard KLS and HKLS. The method establishes that population regression coefficients coincide under (no confounding) and that finite-sample differences follow a zero-mean Gaussian distribution, enabling a statistically sound test. Empirically, KRCD outperforms baselines in NSE and competitive baselines in multi-environment settings, while delivering superior computational efficiency and robustness to kernel choices. This approach broadens confounding detection to nonlinear regimes without requiring instruments or multiple environments, with practical implications for causal inference in observational studies.

Abstract

Detecting unobserved confounders is crucial for reliable causal inference in observational studies. Existing methods require either linearity assumptions or multiple heterogeneous environments, limiting applicability to nonlinear single-environment settings. To bridge this gap, we propose Kernel Regression Confounder Detection (KRCD), a novel method for detecting unobserved confounding in nonlinear observational data under single-environment conditions. KRCD leverages reproducing kernel Hilbert spaces to model complex dependencies. By comparing standard and higherorder kernel regressions, we derive a test statistic whose significant deviation from zero indicates unobserved confounding. Theoretically, we prove two key results: First, in infinite samples, regression coefficients coincide if and only if no unobserved confounders exist. Second, finite-sample differences converge to zero-mean Gaussian distributions with tractable variance. Extensive experiments on synthetic benchmarks and the Twins dataset demonstrate that KRCD not only outperforms existing baselines but also achieves superior computational efficiency.
Paper Structure (46 sections, 8 theorems, 59 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 46 sections, 8 theorems, 59 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Detection of Unobserved Confounders in Single-Environment Settings
  4. Detection of Unobserved Confounders in Multi-Environment Settings
  5. Kernelized Causal Inference
  6. Preliminaries and Problem Setup
  7. Previous Problem Setup
  8. Our Problem: Detecting Unobserved Confounders in Nonlinear Single-Environment (NSE)
  9. Kernel Regression for Confounder Detection
  10. Motivation
  11. Representer for Functional Regression
  12. Theoretical Foundation
  13. Hypothesis Testing with Finite Samples
  14. Complexity Analysis
  15. Experiments
  16. ...and 31 more sections

Key Result

Theorem 1

Let $X \in \mathbb{R}^{N \times d}$ be the design matrix with rank $r < N$, and let $\{\phi_i\}_{i=1}^P$ ($r \leq P < N$) form a basis for a $P$-dimensional subspace $\mathcal{S}_P \subseteq \mathcal{H}_k$ (RKHS). Consider the regularized empirical risk minimization problem: with convex loss $\ell$, convex non-decreasing regularizer $\Omega$, and $\xi > 0$. Then the minimizer $\hat{g}$ admits the

Figures (3)

  • Figure 1: Workflow for Proposed Method (KRCD).
  • Figure 2: KRCD Performance in Nonlinear Single-Environment---(a) ROC Curves across Confounding Strengths. (b) Detection Rate of KRCD, ROCD, and ME-ICM. (c) Detection Rate across Confounding Strengths and Sample Sizes.
  • Figure 3: (a) Detection Rate of KRCD and ME-ICM in Nonlinear Multi-Environment. (b) Detection Rate of KRCD and CNF in Nonlinear Multi-Environment. (c) Performance of Polynomial-KRCD and Gaussian-KRCD.

Theorems & Definitions (13)

  • Remark 1: Distinction from Previous Problem Setups
  • Theorem 1: Variant of Representer Theorem
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Theorem 4: Asymptotic Distribution of $\hat{\boldsymbol{\delta}}$
  • Theorem 5: Variant of Representer Theorem
  • Proof
  • Theorem 6
  • Proof
  • ...and 3 more