Distributional Balancing for Causal Inference: A Unified Framework via Characteristic Function Distance

Weighting methods are essential tools for estimating causal effects in observational studies, with the goal of balancing pre-treatment covariates across treatment groups. Traditional approaches pursue this objective indirectly, for example, via inverse propensity score weighting or by matching a finite number of covariate moments, and therefore do not guarantee balance of the full joint covariate distributions. Recently, distributional balancing methods have emerged as robust, nonparametric alternatives that directly target alignment of entire covariate distributions, but they lack a unified framework, formal theoretical guarantees, and valid inferential procedures. We introduce a unified framework for nonparametric distributional balancing based on the characteristic function distance (CFD) and show that widely used discrepancy measures, including the maximum mean discrepancy and energy distance, arise as special cases. Our theoretical analysis establishes conditions under which the resulting CFD-based weighting estimator achieves $\sqrt{n}$-consistency. Since the standard bootstrap may fail for this estimator, we propose subsampling as a valid alternative for inference. We further extend our approach to an instrumental variable setting to address potential unmeasured confounding. Finally, we evaluate the performance of our method through simulation studies and a real-world application, where the proposed estimator performs well and exhibits results consistent with our theoretical predictions.

💡 Research Summary

This paper addresses a central challenge in causal inference from observational data: achieving balance of the full pre‑treatment covariate distribution across treatment groups. Traditional weighting methods—such as inverse‑probability weighting (IPW) and covariance‑balancing propensity scores—rely either on correctly specified propensity‑score models or on balancing a finite set of moments. Both approaches can leave residual confounding because they do not guarantee alignment of the entire joint distribution of covariates.

The authors propose a unified, non‑parametric framework based on the characteristic function distance (CFD). For two probability distributions (P) and (Q) on (\mathbb{R}^d) and a non‑negative density function (\omega), the CFD is defined as
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