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Nonparametric Identification and Inference for Counterfactual Distributions with Confounding

Jianle Sun, Kun Zhang

Abstract

We propose nonparametric identification and semiparametric estimation of joint potential outcome distributions in the presence of confounding. First, in settings with observed confounding, we derive tighter, covariate-informed bounds on the joint distribution by leveraging conditional copulas. To overcome the non-differentiability of bounding min/max operators, we establish the asymptotic properties for both a direct estimator with polynomial margin condition and a smooth approximation with log-sum-exp operator, facilitating valid inference for individual-level effects under the canonical rank-preserving assumption. Second, we tackle the challenge of unmeasured confounding by introducing a causal representation learning framework. By utilizing instrumental variables, we prove the nonparametric identifiability of the latent confounding subspace under injectivity and completeness conditions. We develop a ``triple machine learning" estimator that employs cross-fitting scheme to sequentially handle the learned representation, nuisance parameters, and target functional. We characterize the asymptotic distribution with variance inflation induced by representation learning error, and provide conditions for semiparametric efficiency. We also propose a practical VAE-based algorithm for confounding representation learning. Simulations and real-world analysis validate the effectiveness of proposed methods. By bridging classical semiparametric theory with modern representation learning, this work provides a robust statistical foundation for distributional and counterfactual inference in complex causal systems.

Nonparametric Identification and Inference for Counterfactual Distributions with Confounding

Abstract

We propose nonparametric identification and semiparametric estimation of joint potential outcome distributions in the presence of confounding. First, in settings with observed confounding, we derive tighter, covariate-informed bounds on the joint distribution by leveraging conditional copulas. To overcome the non-differentiability of bounding min/max operators, we establish the asymptotic properties for both a direct estimator with polynomial margin condition and a smooth approximation with log-sum-exp operator, facilitating valid inference for individual-level effects under the canonical rank-preserving assumption. Second, we tackle the challenge of unmeasured confounding by introducing a causal representation learning framework. By utilizing instrumental variables, we prove the nonparametric identifiability of the latent confounding subspace under injectivity and completeness conditions. We develop a ``triple machine learning" estimator that employs cross-fitting scheme to sequentially handle the learned representation, nuisance parameters, and target functional. We characterize the asymptotic distribution with variance inflation induced by representation learning error, and provide conditions for semiparametric efficiency. We also propose a practical VAE-based algorithm for confounding representation learning. Simulations and real-world analysis validate the effectiveness of proposed methods. By bridging classical semiparametric theory with modern representation learning, this work provides a robust statistical foundation for distributional and counterfactual inference in complex causal systems.
Paper Structure (54 sections, 6 theorems, 145 equations, 6 figures)

This paper contains 54 sections, 6 theorems, 145 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Bound joint counterfactuals with all confounders observed via conditional copulas
  3. Identification
  4. Identification of counterfactual marginals
  5. Conditional Fréchet–Hoeffding bounds on counterfactual joint distributions
  6. Estimation and Inference
  7. Direct Estimation under a Margin Condition
  8. Smooth Approximation via Log-Sum-Exp
  9. Counterfactual marginals with unmeasured confounding: Triple Machine Learning via a Representation Learning based IV Approach
  10. Identification
  11. Estimation and Inference
  12. Standard estimators with observed confounding
  13. Triple Machine Learning estimator
  14. 1. Consistency (double robustness)
  15. 2. Asymptotic Normality
  16. ...and 39 more sections

Key Result

Theorem 1

Assume the standard causal identification conditions

Figures (6)

  • Figure 1: Causal and latent structure underlying the IV-based representation-learning model with Exogenous Noises. $\varepsilon_C$ and $\varepsilon_S$ are the independent, exogenous noise sources. $Z_C$ (pure confounder) is defined by $\varepsilon_C$. $Z_S$ (IV-related latents) is defined by $Z_C$, $S$, and $\varepsilon_S$. The absence of direct edges $S\to Y$ and $Z_S\to Y$ satisfies the Exclusion Restriction.
  • Figure 2: Estimations on the bounds of joint distribution of potential outcomes.
  • Figure 3: Representation learning quality, demonstrated by correlation between true $Z_C$ and learned $\hat{Z}_C$, and the dependence between learned $\hat{Z}_C$ and instrument $S$.
  • Figure 4: Simulation results of ATE estimation with unmeasured confounding. We evaluate 2SLS baseline with different estimators (outcome regression, IPW, and double robust) within triple machine learning (TML) framework.
  • Figure 5: Dose-response curve estimation
  • ...and 1 more figures

Theorems & Definitions (20)

  • Definition 1: Nuisance Functions and Distributions
  • Theorem 1: Identification of counterfactual marginals
  • Theorem 2
  • Theorem 3: Asymptotic properties of margin - direct estimator
  • Remark 1
  • Theorem 4: Properties of the smooth log-sum-exp estimator
  • Remark 2: bias-variance trade-off in limit as $t\to\infty$
  • Remark 3: Lower bound
  • Theorem 5: Identification
  • Remark 4: Completeness v.s. Variability
  • ...and 10 more