Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction (2105.04544v7)

Abstract: We address the problem of causal effect estimation in the presence of unobserved confounding, but where proxies for the latent confounder(s) are observed. We propose two kernel-based methods for nonlinear causal effect estimation in this setting: (a) a two-stage regression approach, and (b) a maximum moment restriction approach. We focus on the proximal causal learning setting, but our methods can be used to solve a wider class of inverse problems characterised by a Fredholm integral equation. In particular, we provide a unifying view of two-stage and moment restriction approaches for solving this problem in a nonlinear setting. We provide consistency guarantees for each algorithm, and we demonstrate these approaches achieve competitive results on synthetic data and data simulating a real-world task. In particular, our approach outperforms earlier methods that are not suited to leveraging proxy variables.

• The paper presents two kernel-based approaches, KPV and PMMR, for estimating causal effects in the presence of unobserved confounders using available proxy variables.
• KPV employs a two-stage regression process that first captures proxy variable distributions and then regresses outcomes, providing consistency in nonparametric settings.
• PMMR leverages a single-stage moment restriction within an RKHS framework, demonstrating superior performance in simulations and empirical applications compared to traditional methods.

Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction

The paper "Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction" presents a sophisticated exploration of estimating causal effects in the presence of unobserved confounders. The methodology pivots on the availability of proxy variables related to these confounders, and leverages two innovative kernel-based approaches for robust estimation: (a) a two-stage regression approach termed Kernel Proxy Variable (KPV) and (b) a Proxy Maximum Moment Restriction (PMMR) method.

Causal Estimation Challenges

In observational studies, unobserved confounders present a significant barrier to accurately estimating causal effects. Traditional methods often rely on the assumption of no unmeasured confounding or employ instrumental variables that are strictly independent of the confounders. The authors tackle the challenge through a proximal causal inference framework, which exploits proxy variables that are observed and carry information about the latent confounders. This paper, therefore, extends the methodological capabilities for causal effect estimation to wider contexts, where such proxies are available, but confounders remain unmeasured.

Kernel-based Methodologies

The paper's kernel-based approaches allow for handling the high-dimensional, nonlinear relationships often encountered in causal inference tasks:

1. Kernel Proxy Variable (KPV) Approach: This two-stage method involves:
   • First, estimating conditional mean embeddings of proxy distributions, which capture the confounder information.
   • Second, learning the causal effect function by regressing outcomes on these embeddings. The authors provide consistency guarantees for this algorithm, underscoring its reliability.
2. Proxy Maximum Moment Restriction (PMMR) Method:
   • PMMR leverages a closed-form solution for the conditional moment restriction, framed within an RKHS.
   • This allows PMMR to directly estimate causal effects in a single-stage process, minimizing a moment criterion.

These kernel approaches improve upon previous methods by not assuming parametric forms or linear relationships, which enhances their applicability across various settings.

Theoretical Foundations and Empirical Results

The theoretical foundation is rigorously developed, ensuring these approaches are mathematically sound with proofs of consistency under relevant conditions, including regularity and completeness assumptions. Empirically, both KPV and PMMR show superior performance in simulations with nonlinear and complex data generation processes, as well as in real-world datasets such as the effect of grade retention on cognitive outcomes and the impact of legalized abortion on crime rates. They outperform traditional approaches and recent nonparametric methods that do not fully utilize proxy information.

Implications and Future Directions

The implications of this work are manifold. Practically, these methods provide researchers with robust tools for causal inference in complex scenarios, broadening the spectrum of applications in fields like epidemiology, social sciences, and economics, where proxy variables are frequently encountered. Theoretically, this work paves the way for further exploration into proxy-assisted causal estimation, particularly in scenarios where the completeness assumptions might also be tested or generalized.

Future research could focus on enhancing computational efficiency, especially when dealing with large-scale data, and exploring hybrid models that combine kernel methods with neural network architectures. Further exploration of the theoretical boundaries of these approaches, particularly regarding the completeness and identifiability under minimal assumptions, would also enrich causal inference literature.

In conclusion, the approaches delineated in this paper represent a notable advancement in causal inference, integrating the flexibility and capacity of kernel methods with a precise focus on overcoming unobserved confounder challenges through proxy variables.