Identifying Causal Effects Using a Single Proxy Variable
| Authors | Silvan Vollmer et al. |
| Year | 2026 |
| Field | Statistics / ML |
| arXiv | 2604.09135 |
| Download | |
| Categories | stat.ML, cs.LG, stat.ME |
Abstract
Unobserved confounding is a key challenge when estimating causal effects from a treatment on an outcome in scientific applications. 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 a completeness assumption on this mechanism, which we call Single Proxy Identifiability of Causal Effects or simply SPICE, we prove that causal effects are identifiable. We extend the proxy-based causal identifiability results by Kuroki and Pearl (2014); Pearl (2010) to higher dimensions, 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.
Engineering Breakdown
Plain English
This paper solves the problem of estimating causal effects when there are unobserved confounding variables—hidden factors that influence both the treatment and outcome, making it impossible to determine true causality. The authors show that if you have access to a single proxy variable (which could be multi-dimensional) that deterministically relates to the unobserved confounder, and you know the mechanism generating that proxy, you can recover the causal effect under a mathematical completeness condition they call SPICE. They extend prior work by Kuroki and Pearl to handle higher-dimensional proxies and more flexible functional relationships, then develop SPICE-Net, a neural network framework to estimate these causal effects in practice.
Core Technical Contribution
The core novelty is the SPICE (Single Proxy Identifiability of Causal Effects) theoretical framework, which rigorously characterizes when a single proxy variable is sufficient to identify causal effects despite unobserved confounding. The key insight is that completeness of the mapping from confounder to proxy—roughly, that the proxy reveals all the confounding information—makes the system identifiable, even with just one proxy. This extends prior identifiability results that required stronger assumptions or multiple proxies, and generalizes to multi-dimensional proxies and non-linear mechanisms. The authors pair this theory with SPICE-Net, a practical neural network estimator that learns to leverage the proxy structure without directly observing the confounder.
How It Works
The method operates in two stages: first, the theoretical identifiability analysis uses the assumed generative mechanism (the known relationship between hidden confounder U and observed proxy W) to construct an estimating equation that does not depend on U directly. Given treatment T, outcome Y, proxy W, and the known proxy mechanism, the completeness assumption guarantees that you can uniquely solve for the causal effect parameters. Second, SPICE-Net is a neural network architecture that implements this identification strategy: it learns to invert or exploit the proxy mechanism to reconstruct information about the confounder, then uses that to adjust the treatment-outcome relationship. The network takes as input {T, Y, W} and the known mechanism parameters, and outputs estimates of the causal effect (e.g., ATE, conditional treatment effects), training on synthetic or observational data where the true causal effect is known or can be validated via simulation.
Production Impact
In production observational data pipelines—such as medical studies, economics, or marketing—this approach directly addresses a critical real-world constraint: you often know the proxy mechanism (e.g., if the confounder is unobserved skill and the proxy is a test score, you may know the relationship mathematically) but never directly measure the confounder itself. Adopting SPICE-Net would allow teams to estimate treatment effects with formal identifiability guarantees rather than relying on unmeasured confounder bounds or sensitivity analyses. Computationally, it requires training a neural network on observational data, which is modest; the main overhead is upfront validation that the completeness assumption holds and careful specification of the proxy mechanism. Integration complexity is moderate—you need domain expertise to specify the proxy mechanism correctly, and any misspecification breaks identifiability, so this requires close collaboration between domain scientists and ML engineers.
Limitations and When Not to Use This
The method critically assumes you know the exact mechanism generating the proxy from the confounder; if this mechanism is misspecified or unknown, identifiability is lost and estimates become biased. The completeness assumption is strong and unverifiable from data alone—you must argue it holds based on domain knowledge, and it may fail in high-dimensional confounding scenarios. The paper does not address model selection, multiple testing, or finite-sample inference, so practitioners must still implement confidence interval estimation and multiple hypothesis correction independently. Extension to settings with multiple confounders, time-varying treatments, or feedback loops remains open; the current framework is designed for simple causal graphs with a single confounder.
Research Context
This work directly extends the proxy variable causal identifiability literature initiated by Pearl (2010) and Kuroki & Pearl (2014), which showed that proxy variables can recover point-identified causal effects under certain conditions. The contribution is moving from restrictive linear models or single-dimensional proxies to general non-linear mechanisms and multi-dimensional proxies, making the theory applicable to richer real-world scenarios. SPICE-Net continues a broader trend of pairing causal identification theory with neural network estimators (similar to work on causal representation learning and neural causal models), enabling scalability to complex data. This opens new directions in deconfounding observational studies and could inspire extensions to multiple confounders, unknown proxy mechanisms, or semi-parametric estimation.
:::tip Subscribe Get weekly breakdowns of papers like this in AI Letters - the newsletter for engineers building production AI systems. :::