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Partial Identification of Causal Effects Using Proxy Variables

This paper proposes partial identification methods that leverage proxy variables to derive bounds on causal effects in the presence of unmeasured confounding, circumventing the need for empirically untestable completeness conditions required for point identification.

Original authors: AmirEmad Ghassami, Yunshu Zhang, Ilya Shpitser, Eric Tchetgen Tchetgen

Published 2026-08-06

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CC BY 4.0

Original authors: AmirEmad Ghassami, Yunshu Zhang, Ilya Shpitser, Eric Tchetgen Tchetgen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery: Did a specific action cause a specific result? In the world of science, this is called figuring out "causal effects." Usually, detectives rely on a perfect crime scene where they can see every single clue. In statistics, this means having a complete list of every factor that could influence both the action and the result. If you have that list, you can confidently say, "This caused that."

But real life is messy. Often, there are "hidden villains"—factors we can't see or measure—that mess up the clues. These are called "unobserved confounders." Maybe a person's hidden stress level makes them both more likely to take a medicine and more likely to get sick. If you don't know about the stress, you might wrongly blame the medicine. For a long time, scientists thought that if you couldn't see the hidden villain, you were stuck. You couldn't solve the mystery.

Then, a new idea called "proximal causal inference" arrived. It's like saying, "Okay, we can't see the hidden villain, but maybe we have a shadow or a footprint nearby that tells us something about them." These "footprints" are called proxy variables. If the shadow is long enough and detailed enough, we can use it to reconstruct the villain's shape and solve the case. However, there was a catch: to use this method perfectly, the shadow had to be a perfect reflection of the villain. If the shadow was blurry or incomplete, the whole method would break down, and the mystery would remain unsolved.

This paper is about what happens when the shadow isn't perfect. The authors, a team of statisticians and computer scientists, realized that we don't always need a perfect shadow to get some answers. Instead of trying to find the exact shape of the hidden villain (which might be impossible), they developed a new way to draw a box around the possible answers. They show that even with a blurry shadow, we can still say, "The answer is definitely somewhere between this number and that number." They don't need the perfect "bridge" to cross the gap; they just need to build a sturdy fence around the truth.

The team tested their new "fence-building" method in two ways. First, they created fake worlds in a computer simulation to see how well their math worked. They found that their method was very good at catching the true answer inside the box, even when the information was imperfect. In fact, in many of their fake scenarios, the box they built was surprisingly tight, giving a very clear range of possibilities. Second, they took their method to a real-world medical mystery involving critically ill patients in an intensive care unit. They wanted to know if a specific procedure (right heart catheterization) helped or hurt patients. Using their new method with data from over 5,000 patients, they found that the procedure likely shortened the time patients survived. Their results matched what other scientists had found using different, more demanding methods, proving that their "fence" approach works in the real world.

The paper doesn't claim to have solved the mystery of every hidden villain forever. Instead, it offers a powerful new tool for when the clues are incomplete. It tells us that even when we can't pinpoint the exact answer, we can still narrow down the possibilities enough to make smart, informed decisions. It turns a "we don't know" into a "we know it's somewhere in this range," which is often exactly what we need to move forward.

Technical Summary: Partial Identification of Causal Effects Using Proxy Variables

Problem Statement
The evaluation of causal effects from observational data often relies on the assumption of conditional exchangeability, which requires that all common causes of the treatment and outcome are observed. In the presence of unmeasured confounders, this assumption is violated. The proximal causal inference framework, introduced by Tchetgen Tchetgen et al. (2020), addresses this by utilizing proxy variables for unobserved confounders to achieve point identification. However, this framework typically requires a "completeness condition," ensuring that proxy variables capture all sources of variation in the unobserved confounders. This condition is non-testable and often fails in practical settings, particularly when only one type of proxy is available or when proxies are insufficiently rich. Consequently, the identification of the necessary "bridge functions" (solutions to integral equations involving unobserved variables) becomes impossible, rendering point identification of causal effects infeasible.

Methodology
This paper proposes a partial identification framework that does not rely on completeness conditions or the point identification of bridge functions. The authors establish bounds on causal parameters (specifically the Average Treatment Effect (ATE) and the Effect of Treatment on the Treated (ETT)) using proxy variables under weaker assumptions.

The methodology is developed across three primary settings:

Single Proxy Settings:
- Outcome Confounding Proxy ( $W$ ): Assumes $W \perp \perp A \mid \{X, U\}$ (independent of treatment given confounders) and the existence of a non-negative bridge function $h$ linking the potential outcome mean to $W$ . Without completeness, the authors derive bounds for conditional and marginal potential outcome means based on the ratio of conditional probabilities of the proxy.
- Treatment Confounding Proxy ( $Z$ ): Assumes $Z \perp \perp Y \mid \{A, X, U\}$ (independent of outcome given treatment and confounders) and the existence of a bridge function $q$ relating the likelihood ratio of the unobserved confounder to $Z$ . Bounds are derived using the conditional expectation of the outcome given the proxy.
Two Independent Invalid Proxies: The framework is extended to settings with two proxies ( $W$ and $Z$ ) that are conditionally independent given confounders but may not satisfy strict exclusion restrictions (i.e., they may be directly associated with treatment or outcome). This allows for valid bounds even when individual proxies are "invalid" in the traditional sense.
Unobserved Mediators: The approach is adapted to mediation analysis and front-door models where the mediator is unobserved but a proxy is available. The authors provide partial identification bounds for Natural Indirect Effects (NIE) and Natural Direct Effects (NDE), as well as the ATE in front-door settings, without requiring point identification of the mediator's distribution.

Inference and Estimation
The derived bounds are non-smooth functionals of the observed data distribution (involving $\min$ and $\max$ operators), which complicates standard statistical inference. To address this, the authors:

Employ LogSumExp (LSE) approximations to construct smooth lower and upper bounds.
Derive the Efficient Influence Functions (EIFs) for these smooth bound functionals.
Utilize these EIFs to enable one-step estimators with analytic variance estimation.
Note that for regular plug-in estimators, standard bootstrap confidence intervals remain valid.

Key Results and Contributions

Relaxation of Completeness: The paper demonstrates that causal effects can be partially identified even when the completeness condition fails, obviating the need to solve integral equations for bridge functions.
Generalization of Proxy Types: The framework accommodates settings with single proxies, two independent proxies (even if "invalid"), and unobserved mediators, broadening the applicability of proximal causal inference.
Inference Framework: By deriving EIFs for smooth approximations of the bounds, the paper provides a rigorous basis for analytic standard errors and confidence intervals, moving beyond purely simulation-based inference.
Simulation Performance: Simulation studies indicate that the proposed methods yield valid 95% confidence intervals when assumptions hold. Notably, the "Z-based" method (using a treatment confounding proxy) often produces tighter bounds than the "W-based" method. The simulations also suggest robustness to slight violations of the bridge function existence assumption, with high coverage rates even when the number of proxy states is smaller than the unobserved confounder states.
Empirical Application: The method is applied to the SUPPORT dataset to evaluate the effect of Right Heart Catheterization (RHC) on survival. Using two different proxy variables (PaFI and PaCO2), the study finds negative causal effects of RHC on survival, consistent with previous literature that utilized point identification methods.

Significance
The paper claims that its primary contribution is establishing that proxies can be leveraged to obtain bounds on causal effects even when available information is insufficient for point identification. By removing the reliance on the untestable completeness condition, the authors argue that the proximal causal inference framework becomes applicable in a wider range of practical settings where only limited proxy information is available. The development of smooth approximations and efficient influence functions further enables robust statistical inference for these partial identification results.

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