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.
Causal inference fundamentally aims to predict how individuals or populations respond to competing interventions, thereby concerning the comparison of potential outcomes under alternative treatment regimes. While classical estimands such as the Average Treatment Effect (ATE) focus on mean differences, many scientifically relevant questions, such as the probability of benefit, quantile effects, or distributional shifts, depend on the entire distributions of potential outcomes. However, researchers typically face a dual hurdle in capturing these distributions. First, in the presence of confounding, even the marginal distributions of Y (1) and Y (0) are generally not identifiable, and existing instrumental variable (IV) approaches often rely on restrictive parametric assumptions or focus only on local effects Angrist et al. (1996), Swanson et al. (2018). Second, even when conditional ignorability holds and marginals are identifiable, the joint distribution (Y (1), Y (0)) remains fundamentally unobservable without additional structural assumptions. This paper bridges these gaps by providing a unified, principled framework for distributional causal inference. Our first contribution addresses the "missing data" problem of the joint distribution under the assumption of no unmeasured confounding. We develop tight, covariate-informed Fréchet-Hoeffding (FH) bounds Nelsen (2006) on the joint distribution under no unmeasured confounding. Leveraging conditional copulas, we show that the sharp upper bound admits a clear structural interpretation as conditional rank preservation (or conditional comonotonicity) Nelsen (2006), a canonical assumption underlying individual treatment effect estimation and counterfactual reasoning Xie et al. (2023), Wu et al. (2025). To move from theory to practice, we address the non-smoothness of these bounds via two complementary paths: a direct estimator under a polynomial margin condition and a smooth log-sum-exp approximation. We further establish their asymptotic properties, enabling valid frequentist inference and confidence intervals for rank-preserving structures Levis et al. (2025).
Our second contribution tackles the more daunting scenario where confounding is unmeasured, and in this scenario even the non-parametric identification for marginal distributions is non-trivial. Inspired by recent advances in causal representation learning Kong et al. (2022), Ng, Blöbaum, Bhandari, Zhang & Kasiviswanathan (2025), Moran & Aragam (2026), we propose a representation learning based framework that leverages IVs to recover latent confounding structures. Under suitable completeness and independence assumptions, we show that the confounding subspace is identified up to an invertible transformation. This allows the learned representation to serve as a valid proxy for unobserved confounders, thereby enabling identification of marginal potential outcome distributions in complex settings.
To implement this framework, we introduce a triple machine learning (TML) procedure that extends double machine learning Chernozhukov et al. (2018), Kennedy (2024) by incorporating an additional cross-fitting stage for representation learning. We rigorously characterize the impact of first-stage representation error on the variance inflation in asymptotic distribution, identifying regimes in which super-convergence of the representation learner yields semiparametric efficiency.
For practical estimation on confounding representations, we propose an Instrumental Variable Variational Autoencoder (IV-VAE) augmented with a Hilbert-Schmidt Independence Criterion (HSIC) penalty Gretton et al. (2005) to ensure that the recovered latent factors are truly exogenous to the instrument, satisfying the core identifying assumptions. Rather than explicitly modeling instrument-dependent latent factors, we adopt a reduced-form design that conditions the decoder directly on the observed instrument, allowing instrument-induced variation to be absorbed while isolating latent confounding structure.
The remainder of the paper is organized as follows. Section 2 details the identification of rank-preserving bounds via conditional copulas and presents the asymptotic theory for the proposed estimators. Section 3 introduces the representation learning framework for unmeasured confounding, derives the properties of the triple machine learning estimator, and proposes an effective VAE-based learning approach. Section 4 discusses the synthesis of these methods. Section 5 provides simulation results. Section 6 applies our proposed methods to analyze the demand for cigarettes in US. Proofs and technical details are deferred to the Appendix.
2 Bound joint counterfactuals with all confounders observed via conditional copulas When there is no unobserved confounding, the identification of counterfactual marginals F Y (a) is easy. We will briefly review the identification results for the marginal distributions of potential outcomes and discuss lever
This content is AI-processed based on open access ArXiv data.