Identifying Causal Effects Using a Single Proxy Variable

Silvan Vollmer; Niklas Pfister; Sebastian Weichwald

stat.MLApr 10, 2026

Identifying Causal Effects Using a Single Proxy Variable

Authors: Silvan Vollmer, Niklas Pfister, Sebastian Weichwald

Organizations: Department of Mathematical Sciences University of Copenhagen Copenhagen, Denmark · Lakera AI Zurich, Switzerland

Abstract

Unobserved confounding is a key challenge when estimating causal effects from a treatment on an outcome. In this work, we assume that we observe a single, potentially multi-dimensional proxy variable of the unobserved confounder and that we know the mechanism that generates the proxy from the confounder. Under an assumption called Single Proxy Identifiability of Causal Effects or simply SPICE, we prove that this error mechanism is complete and causal effects are identifiable. We extend the proxy-based causal identifiability results by Kuroki and Pearl (2014); Pearl (2010) to multi-dimensional continuous settings, more flexible functional relationships and a broader class of distributions. Further, we develop a neural network based estimation framework, SPICE-Net, to estimate causal effects, which is applicable to both discrete and continuous treatments.

Causal Effect Estimation

Causal effect estimation measures how outcomes would change under interventions, using experiments or observational data and explicit assumptions.

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Proximal Balancing for Causal Effect Estimation under Unmeasured Confounding

Yonghan Jung

School of Information Sciences University of Illinois Urbana-Champaign

Causal Effect Estimation Proximal Causal Inference Causal Inference Causal Inference with Unobserved Confounding

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Estimating causal effects from observational data is central to science and policy, but the effects are not identified when confounders are unmeasured. Proximal causal inference addresses this problem with proxies of the unmeasured confounders. However, existing proxy-based approaches either designate proxy roles and solve an inverse problem, which is ill-posed and hard to estimate with high-dimensional proxies, or use a latent-variable model, which assumes that the learned latent variable matches the hidden confounder and leaves bias when it does not. To address these challenges, we introduce proximal balancing. It carries the classical idea of covariate balancing to confounders that are observed only through proxies: it learns a low-dimensional summary of the covariates and proxies that makes the treatment groups comparable, and then adjusts for this summary. It needs no designated proxy roles, inverse problem, or latent model. We give identification theory, finite-sample guarantees, and a practical algorithm, PROBE. We demonstrate the method on low-dimensional, high-dimensional, and image proxies and on real-world data.

Abstract

Causal Effect Estimation

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