Sensitivity Analysis to Unobserved Confounding with Copula-based Normalizing Flows

2025-08-12

AI Summary

To address bias from unobserved confounding in causal inference, this paper proposes the ρ-GNF model. Methodologically, it is the first to integrate Gaussian copulas with normalizing flows to construct an interpretable density estimation framework for causal graphs. It introduces a sensitivity parameter ρ that quantifies the strength of non-causal association between exposure and outcome, yielding a ρ-curve—i.e., the average causal effect (ACE) as a function of ρ—to visually assess effect robustness and identify the critical confounding threshold at which ACE vanishes. Furthermore, a Bayesian extension enables posterior inference of ACE and credible interval estimation. Empirical evaluation on synthetic and real-world datasets demonstrates that ρ-GNF accurately estimates causal effects across varying confounding strengths and substantially enhances the interpretability and practical utility of sensitivity analysis.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We propose a novel method for sensitivity analysis to unobserved confounding in causal inference. The method builds on a copula-based causal graphical normalizing flow that we term $ρ$-GNF, where $ρin [-1,+1]$ is the sensitivity parameter. The parameter represents the non-causal association between exposure and outcome due to unobserved confounding, which is modeled as a Gaussian copula. In other words, the $ρ$-GNF enables scholars to estimate the average causal effect (ACE) as a function of $ρ$, accounting for various confounding strengths. The output of the $ρ$-GNF is what we term the $ρ_{curve}$, which provides the bounds for the ACE given an interval of assumed $ρ$ values. The $ρ_{curve}$ also enables scholars to identify the confounding strength required to nullify the ACE. We also propose a Bayesian version of our sensitivity analysis method. Assuming a prior over the sensitivity parameter $ρ$ enables us to derive the posterior distribution over the ACE, which enables us to derive credible intervals. Finally, leveraging on experiments from simulated and real-world data, we show the benefits of our sensitivity analysis method.

Problem

• Estimates causal effects under unobserved confounding.
• Models confounding strength using Gaussian copula.
• Provides Bayesian sensitivity analysis for causal inference.

Innovation

• Copula-based normalizing flows for sensitivity analysis
• Bayesian method for posterior ACE distribution
• ρ-GNF models unobserved confounding via Gaussian copula