Proximal Balancing for Causal Effect Estimation under Unmeasured Confounding

Yonghan Jung

stat.MLSep 30, 2026

Proximal Balancing for Causal Effect Estimation under Unmeasured Confounding

Authors: Yonghan Jung

Organizations: School of Information Sciences University of Illinois Urbana-Champaign

Abstract

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.

Figures & tables

Figure 1: (a) The proxy

W

, split into

(W_{j},W_{-j})

. (b) A higher-resolution view of (a), showing an allowed structure: the held-out block

W_{j}

affects

Y

but not

A

. (c) Proximal causal inference with designated proxies

Z^{\mathrm{tr}}

and

Z^{\mathrm{out}}

Method	Roles	Inverse	Latent
Proximal inference	$\times$	$\times$	$\checkmark$
Single-proxy control	$\times$	$\times$	$\checkmark$
CEVAE	$\checkmark$	$\checkmark$	$\times$
Saha et al.,	$\checkmark$	$\checkmark$	$\times$
Proximal balancing	$\checkmark$	$\checkmark$	$\checkmark$

Table 1: Requirements of proxy-based methods; a check means not needed. Roles: designated proxy roles. Inverse: an inverse problem. Latent: a latent model of

U

Figure 2: PROBE learns, audits, and aggregates across held-out splits.

Figure 3: Estimate minus the true effect on a symmetric log scale; each dot is one data set. Filled and hollow dots show methods that need proxy roles with correct and wrong roles; on the semi-synthetic data, single-proxy control is used as an ATE estimator.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Causal structures compatible with proximal balancing in panels (a)–(c) and a standard two-proxy proximal structure in panel (d).

Figure 5: Median aggregation over five singleton proxy splits.

Figure 6: Detailed version of the left of Fig. 3 for the original

20

data sets per setting, with every method on its own row. KSPC (

\dagger

) was run after the protocol was frozen.

Figure 7: Kernel single-proxy benchmark of Xu and Gretton, (2025) with a binary treatment; the effect is exactly

1

. Estimates minus the effect over

20

replicates, with the MAE on the right.

Setting	What the learner sees	Mean error	p90	Oracle gap	Status
SCM-1	$8$ coordinates	0.020	0.033	0.022	12/12 gates
SCM-2	$88$ coordinates	0.017	0.037	0.019	12/12 gates
SCM-3	$1{,}312$ coordinates	0.014	0.035	0.015	12/12 gates
SCM-4	six $64\times 64$ images	0.018	0.046	0.023	13/13 gates, amended

Appendix

Table 2: Sealed confirmations. Each row is

20

independently generated data sets at

n=24{,}000

Figure 8: Detailed version of the left of Fig. 3 over all

100

data sets per setting, with the MAE of each method on the right; for methods that need proxy roles, the upper number is for the correct roles.

Figure 9: Detailed version of the right of Fig. 3 over

100

Twins,

88

IHDP, and

9

ACIC data sets, with the MAE of each method, and the RHC estimates, where no true effect is known; PROBE’s estimate carries its resampled

95\%

interval, and * marks estimates published by Cui et al., (2024) .

Method	Twins	IHDP	ACIC
PROBE (ours)	0.0077	0.3054	0.2025
CEVAE	0.0894	0.1828	0.2575
Adjust for $X$ and all of $W$	0.0207	0.3927	0.2239
Adjust for $X$	0.0749	0.7221	0.1757
Proximal 2SLS, roles given	0.0156	0.2328	0.1016
Proximal 2SLS, roles swapped	0.2453	1.5856	0.8015

Appendix

Table 3: Mean absolute error on the original replicates in three semi-synthetic studies with known effects; ACIC is exploratory. Proximal 2SLS, single-proxy control, and KSPC require proxy roles; KSPC (

\dagger

) was run after the protocol was frozen. Bold marks the smallest error among the methods other than the oracle.

Figure 10: Three ways to combine the retained splits. Averaging is pulled away from

\tau

by the balanced but invalid split; the median of the largest agreeing group is not.

Figure 11: Coverage of a nominal

95\%

interval around one split, one dot per data set. The interval covers the population effect of the learned representation at the nominal rate and covers

\tau

less often, by the representation gap.

Figure 12: Absolute errors on a log scale. For one split the representation gap

|\tau_{\widehat{Z}}-\tau|

is the population effect of the learned representation, which is exact for the linear representations of SCM-1 to SCM-3 and is not recorded for SCM-4. Aggregation reduces the error of a single split by roughly a factor of three.

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Aug 2, 2026, cs.LGJ/K move · Enter open · S save

Spatiotemporal Proximal Causal Inference under Hidden Confounding and Interference

Omar Faruque, Pavan Raj Ravi, Jianwu Wang

University of Maryland, Baltimore County

Latent Confounders Identifiability Interference

Open paper AskPDF arXiv

Estimating causal effects from real-world spatiotemporal data is challenging due to hidden confounders and interference. Standard causal identification methods assume conditional exchangeability given observed covariates, which fails whenever hidden confounders affect both treatment and outcomes - a common setting in domains such as climate, environmental policy, epidemiology, and regional economics. In this paper, we propose a novel spatiotemporal proximal causal inference framework that extends proximal identification theory to spatiotemporal settings. The proposed method jointly captures local and neighborhood-level confounding information by introducing treatment- and outcome-inducing proxies, and we derive a spatiotemporal outcome confounding bridge function that identifies the potential outcome without requiring direct recovery of the hidden confounder. We establish the identifiability of this bridge function under proxy exclusion restrictions and a spatiotemporal completeness condition, and show that the resulting estimator recovers the outcome through a proximal generalization of the g-computation formula. To operationalize this identification result, we propose a neural architecture that learns proxies via transformer-based spatiotemporal encoders - coupled with a conditional mutual information critic to enforce exclusion restrictions and a moment-matching network to guarantee that the learned bridge function satisfies the underlying identifying equation. We further introduce a stabilized weighting scheme to address treatment support imbalance. Experiments on synthetic datasets demonstrate that our approach achieves comparable performance to baseline causal inference methods, while providing, to our knowledge, the first theoretically grounded outcomes for the hidden confounding in the presence of spatiotemporal interference through a proximal causal inference framework.

Abstract

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