Proximal Causal Inference for Hidden Outcomes

Technical Summary: Proximal Causal Inference for Hidden Outcomes

Problem Statement
Hidden variables remain a central challenge in causal inference, often preventing the identification and estimation of causal targets. While proxy-based approaches have emerged to circumvent hidden structure without relying on strong parametric assumptions, existing literature typically requires at least one unbiased proxy measurement or partial observation of the true outcome. This paper addresses the specific setting where the outcome variable $Y$ is unobserved (hidden), but multiple potentially biased proxies ($W, Z, V$) are available alongside observed confounders $C$ and treatment $A$. The goal is to identify the full-data law and estimate causal effects (specifically the average treatment effect) without assuming the proxies are unbiased or that the true outcome is partially observed.

Methodology
The authors employ a framework based on the reconstruction of latent distributions from observed data, leveraging the eigenvalue-eigenvector structure inherent in multi-way arrays (tensors).

1. Identification Strategy:

   • The authors define a causal Directed Acyclic Graph (DAG) where $A$ influences $Y$, and $C$ confounds the relationship. The hidden outcome $Y$ generates three observed proxies $W, Z, V$.
   • Assumptions: The identification relies on a set of assumptions (4–8) adapted from Hu and Schennach [11], Deaner [5], and Zhou and Tchetgen Tchetgen [28]. Key conditions include:
     • Mutual conditional independence of proxies given the latent outcome, treatment, and confounders ($W, Z, V \perp \perp \mid Y, A, C$).
     • Completeness conditions on the conditional distributions of the proxies, ensuring sufficient variability to recover the latent distribution.
     • A labeling assumption (Assumption 8) to recover the specific values of the latent outcome, either via a known functional of the conditional distribution or by assuming the outcome is ordinal with known support.
   • Under these conditions, the paper establishes that the full-data law $p(A, Y, C, W, Z, V)$ is identified, thereby identifying the potential outcome distribution $p(Y(a))$ and the causal effect $E[Y(1)] - E[Y(0)]$.

2. Estimation Framework:

   • The authors develop influence function-based estimators for the identified causal effects.
   • Observed-Data Influence Function: Since the true outcome $Y$ is unobserved, the full-data influence function cannot be computed directly. The authors construct an observed-data influence function $\phi_{obs}$ by replacing $Y$ with a weighted average over possible values $y$.
   • Weight Construction: The weights $\omega_y(O; \eta)$ are constructed using the observed proxies and nuisance functions (conditional expectations of the proxies given $Y, A, C$). These weights are designed such that their conditional expectation given the true outcome $Y=y$ is 1, and 0 otherwise.
   • Multiple Robustness: The resulting estimator solves a mean-zero estimating equation. It exhibits multiple robustness: consistency is preserved if any one of several specific sets of nuisance functions (involving the treatment propensity score, outcome regression, or proxy regressions) is correctly specified.
   • Asymptotic Properties: Under standard regularity conditions, the estimator is $\sqrt{n}$-consistent and asymptotically normal. Notably, the nuisance functions do not need to be estimated at $\sqrt{n}$ rates individually; the estimator achieves the desired rate provided the product of estimation errors converges at a sufficient rate.

Key Contributions

• First Semiparametric Estimation for Hidden Outcomes without Unbiased Proxies: To the best of the authors' knowledge, this is the first work to establish semiparametric estimation results for hidden outcomes that do not rely on the existence of unbiased proxy measurements or partial observations of the true outcome.
• Influence Function-Based Estimators: The paper develops the first influence function-based estimators in this specific setting, providing a pathway to achieve multiple robustness and desirable efficiency properties.
• Theoretical Guarantees: The work provides formal proofs for the identification of the full-data law and the asymptotic properties (consistency and normality) of the proposed estimators.

Results
The performance of the proposed estimators was evaluated through simulation studies on both discrete (binary) and mixed (continuous confounders and proxies) data-generating processes.

• Bias Reduction: In both scenarios, a "naïve" estimator that substitutes the unobserved outcome with a proxy was found to substantially underestimate the target parameter (biased). The proposed estimator successfully eliminated this bias.
• Efficiency Trade-off: While the proposed estimator achieved unbiasedness, it incurred a higher variance compared to an "oracle" estimator that had direct access to the true outcome $Y$. In the binary case, the scaled variance was approximately 1.7 versus 0.6 for the oracle. In the mixed case, the variance was significantly higher for smaller sample sizes but decreased as sample size increased.
• Robustness: The simulations confirmed the theoretical properties, demonstrating that the estimator performs well even when the latent outcome must be reconstructed from observed proxies.

Significance and Claims
The authors position this work as a significant step forward in proximal causal inference. By relaxing the requirement for unbiased proxies—a strong assumption often difficult to verify in practice—the framework extends the applicability of proxy-based methods to a broader class of problems involving hidden outcomes. The development of influence function-based estimators allows for the application of modern semiparametric techniques (such as double robustness and sample splitting) to settings previously limited by identification constraints or reliance on parametric assumptions. The paper concludes that while challenges remain regarding optimal proxy combination and practical nuisance function estimation in high-dimensional settings, the proposed framework establishes a viable theoretical and practical foundation for causal inference with hidden outcomes using only biased proxies.