# Partial Identification of Causal Effects Using Proxy Variables

arXiv: https://arxiv.org/abs/2304.04374 · HTML: https://mlsanity.com/paper/2304.04374/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 35 references that are in the map.
- First posted on arXiv 10 Apr 2023.

## Builds on (most cited first, 19 of 19)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Direct and Indirect Effects](https://mlsanity.com/paper/1301.2300/) (2013)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Semiparametric proximal causal inference](https://mlsanity.com/paper/2011.08411/) (2020)
- [A confounding bridge approach for double negative control inference on causal effects](https://mlsanity.com/paper/1808.04945/) (2018)
- [Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference](https://mlsanity.com/paper/2104.02929/) (2021)
- [A Selective Review of Negative Control Methods in Epidemiology](https://mlsanity.com/paper/2009.05641/) (2020)
- [Proximal Causal Inference for Complex Longitudinal Studies](https://mlsanity.com/paper/2109.07030/) (2021)
- [Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closedness](https://mlsanity.com/paper/2302.05404/) (2023)
- [The Proximal ID Algorithm](https://mlsanity.com/paper/2108.06818/) (2021)
- [Proximal mediation analysis](https://mlsanity.com/paper/2109.11904/) (2021)
- [Covariate-assisted bounds on causal effects with instrumental variables](https://mlsanity.com/paper/2301.12106/) (2023)
- [A Universal Approximation Result for Difference of log-sum-exp Neural Networks](https://mlsanity.com/paper/1905.08503/) (2019)
- [Causal Inference with Hidden Mediators](https://mlsanity.com/paper/2111.02927/) (2021)
- [Double Negative Control Inference in Test-Negative Design Studies of Vaccine Effectiveness](https://mlsanity.com/paper/2203.12509/) (2022)
- [Single Proxy Control](https://mlsanity.com/paper/2302.06054/) (2023)
- [Identification and Estimation of Causal Peer Effects Using Double Negative Controls for Unmeasured Network Confounding](https://mlsanity.com/paper/2109.01933/) (2021)
- [Proximal Causal Inference without Uniqueness Assumptions](https://mlsanity.com/paper/2303.10134/) (2023)

## Built on by (most cited first, 7 of 7)

- [Causal Reinforcement Learning: A Survey](https://mlsanity.com/paper/2307.01452/) (2023)
- [Causal Discovery via Conditional Independence Testing with Proxy Variables](https://mlsanity.com/paper/2305.05281/) (2023)
- [Predictive Performance Comparison of Decision Policies Under Confounding](https://mlsanity.com/paper/2404.00848/) (2024)
- [Information-Theoretic Causal Bounds under Unmeasured Confounding](https://mlsanity.com/paper/2601.17160/) (2026)
- [Computational Identifiability](https://mlsanity.com/paper/2606.19361/) (2026)
- [Detecting clinician implicit biases in diagnoses using proximal causal inference](https://mlsanity.com/paper/2501.16399/) (2025)
- [Point-Identification of a Robust Predictor Under Latent Shift with Imperfect Proxies](https://mlsanity.com/paper/2603.15158/) (2026)

## Abstract (arXiv)

Proximal causal inference is a framework for evaluating the causal effects in the presence of unmeasured confounding. For point identification, it leverages a pair of proxy variables to identify a bridge function that matches the dependence of potential outcomes or treatment variables on the hidden factors to corresponding functions of observed proxies. Unique identification requires that proxies are sufficiently relevant for hidden factors, a requirement that has previously been formalized as a completeness condition. However, completeness is not empirically testable, and although a bridge function may be well-defined in a given setting, lack of completeness, sometimes manifested by availability of a single type of proxy, may severely limit prospects for identification of a bridge function and thus a causal effect; therefore, potentially restricting the application of the framework. In this paper, we propose partial identification methods that do not require completeness and obviate the need for identification of a bridge function. We establish that proxies can be leveraged to obtain bounds on the causal effect even if available information does not suffice to identify either a bridge function or a corresponding causal effect of interest. Our bounds are non-smooth functionals of the underlying distribution. For inference, we employ LogSumExp approximations that yield smooth lower and upper bounds, and we derive the efficient influence functions of the resulting bound functionals which enable analytic variance estimation, while bootstrap confidence intervals remain available for regular plug-in implementations. We further establish analogous results in related settings where identification hinges upon hidden mediators for which proxies are available, however such proxies are not sufficiently rich for point identification of a bridge function or a corresponding causal effect of interest.