Proximal Causal Learning under Unmeasured Confounding

Ying Tang; Yi Wang

cs.LGOct 3, 2026

Proximal Causal Learning under Unmeasured Confounding

Authors: Ying Tang, Yi Wang

Organizations: Shanghai University of International Business and Economics, Shanghai 201620, China

Abstract

Estimating treatment effects from observational data typically relies on the No Unmeasured Confounding Assumption (NUCA), which rarely holds in practice. Proximal causal learning (PCL) addresses unmeasured confounding via proxy variables, yet existing methods require the proxy variables to be pre-specified. Thus, we propose PCL-U, a framework that learns proxy variables directly from observed covariates. PCL-U uses neural encoders to decompose covariates into treatment-inducing, outcome-inducing, and shared proxies, guided by minimax mutual information objectives, and obtains causal estimates through a practical moment-based risk function. Experiments on benchmarks show that PCL-U matches or outperforms existing baselines. Besides, there are two types of synthetic datasets with varying dimensions and confounding strengths that illustrate that our method maintains stable estimation accuracy.

Figures & tables

Panel of Figure 1 — Figure 1: Proximal causal graph.

Figure 2: Overview of the PCL-U architecture.

Property	IHDP	Jobs	Twins	Synthetic
F	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
CF	$\checkmark$	$\times$	$\checkmark$	$\checkmark$
Treatment	Binary	Binary	Binary	Binary
Number	747	3,212	11,400	1,000
Covariates	25	17	30	12
Train/valid/test	63/27/10	56/24/20	56/24/20	56/24/20

Table 1: Summary of the datasets.

Datasets	IHDP		Jobs		Twins
Methods	PEHE	$\epsilon_{ATE}$	$\mathcal{R}_{pol}$	$\epsilon_{ATT}$	AUC	$\epsilon_{ATE}$
CFR-MMD	$1.202\pm 0.521$	$0.275\pm 0.120$	$0.245\pm 0.015$	$\mathbf{0.064\pm 0.026}$	$0.540\pm 0.013$	$0.013\pm 0.002$
CFR-WASS	$0.804\pm 0.257$	$0.169\pm 0.065$	$0.252\pm 0.015$	$0.087\pm 0.040$	$0.511\pm 0.015$	$0.014\pm 0.003$
DRRL	$2.154\pm 0.881$	$0.500\pm 0.152$	$0.256\pm 0.015$	$0.109\pm 0.043$	$0.856\pm 0.004$	$0.013\pm 0.002$
SITE	$0.888\pm 0.345$	$0.259\pm 0.099$	$0.249\pm 0.009$	$0.067\pm 0.030$	$0.699\pm 0.032$	$0.023\pm 0.009$
ABCEI	$1.059\pm 0.124$	$0.234\pm 0.030$	$0.345\pm 0.006$	$0.108\pm 0.026$	$0.647\pm 0.028$	$0.010\pm 0.003$

Table 2: Out-of-sample results on IHDP, Jobs and Twins.

Table 5

Ablation		IHDP	1-4-4-4	2-4-4-4 (m)	s	h
w/o $\mathcal{L}_{\mathrm{proxy}}$	within-sample	$0.102\pm 0.011$	$0.050\pm 0.018$	$0.089\pm 0.039$	$0.055\pm 0.015$	$0.599\pm 0.212$
w/o $\mathcal{L}_{\mathrm{proxy}}$	out-of-sample	$0.136\pm 0.021$	$0.056\pm 0.021$	$0.092\pm 0.038$	$0.057\pm 0.014$	$0.599\pm 0.213$
w/o $\mathcal{L}_{O}$	within-sample	$0.124\pm 0.014$	$\mathbf{0.032\pm 0.014}$	$0.045\pm 0.014$	$0.032\pm 0.007$	$0.104\pm 0.043$
w/o $\mathcal{L}_{O}$	out-of-sample	$0.171\pm 0.023$	$0.040\pm 0.018$	$0.055\pm 0.016$	$0.034\pm 0.007$	$0.105\pm 0.044$
Full	within-sample	$\mathbf{0.094\pm 0.012}$	$0.035\pm 0.014$	$\mathbf{0.032\pm 0.011}$	$\mathbf{0.021\pm 0.008}$	$\mathbf{0.080\pm 0.039}$
Full	out-of-sample	$\mathbf{0.129\pm 0.019}$	$\mathbf{0.036\pm 0.012}$	$\mathbf{0.037\pm 0.013}$	$\mathbf{0.016\pm 0.006}$	$\mathbf{0.084\pm 0.037}$

Table 5: Ablation study.

Panel of Figure 3 — Figure 3: Sensitivity of PCL-U to $\alpha$ , $\beta$ , and $\gamma$ on IHDP and the synthetic 2-4-4-4 setting.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Datasets	IHDP		Jobs		Twins
Methods	PEHE	$\epsilon_{ATE}$	$\mathcal{R}_{pol}$	$\epsilon_{ATT}$	AUC	$\epsilon_{ATE}$
CFR-MMD	$0.692\pm 0.090$	$0.240\pm 0.088$	$0.223\pm 0.004$	$0.039\pm 0.018$	$0.549\pm 0.010$	$0.017\pm 0.001$
CFR-WASS	$0.508\pm 0.046$	$0.155\pm 0.054$	$0.221\pm 0.004$	$0.063\pm 0.023$	$0.516\pm 0.013$	$0.017\pm 0.000$
DRRL	$1.386\pm 0.247$	$0.553\pm 0.138$	$0.233\pm 0.003$	$0.054\pm 0.020$	$0.856\pm 0.003$	$0.014\pm 0.001$
SITE	$0.623\pm 0.113$	$0.329\pm 0.138$	$0.223\pm 0.004$	$0.045\pm 0.018$	$0.709\pm 0.033$	$0.024\pm 0.009$
ABCEI	$0.872\pm 0.055$	$0.220\pm 0.027$	$0.217\pm 0.001$	$0.066\pm 0.026$	$0.648\pm 0.027$	$0.007\pm 0.003$

Appendix

Table 6: Within-sample results on IHDP, Jobs and Twins.

Methods	1-4-4-4	2-4-4-4 (m)	8-4-4-4
CFR-MMD	$0.074\pm 0.024$	$0.097\pm 0.023$	$0.344\pm 0.074$
CFR-WASS	$0.046\pm 0.013$	$0.097\pm 0.023$	$0.394\pm 0.061$
DRRL	$0.066\pm 0.014$	$0.107\pm 0.031$	$0.434\pm 0.072$
SITE	$0.119\pm 0.029$	$0.172\pm 0.050$	$0.510\pm 0.062$
ABCEI	$0.060\pm 0.018$	$0.072\pm 0.021$	$0.345\pm 0.067$
CBRE	$0.029\pm 0.011$	$0.046\pm 0.019$	$0.257\pm 0.055$

Appendix

Table 7: Within-sample results on dimension datasets (

d_{u}

d_{z}

d_{x}

d_{w}

Methods	s	m (2-4-4-4)	h
CFR-MMD	$0.031\pm 0.012$	$0.097\pm 0.023$	$0.362\pm 0.068$
CFR-WASS	$0.086\pm 0.033$	$0.097\pm 0.023$	$0.086\pm 0.033$
DRRL	$0.031\pm 0.007$	$0.107\pm 0.031$	$0.338\pm 0.055$
SITE	$0.067\pm 0.025$	$0.172\pm 0.050$	$0.444\pm 0.116$
ABCEI	$0.021\pm 0.008$	$0.072\pm 0.021$	$0.299\pm 0.109$
CBRE	$\mathbf{0.008\pm 0.003}$	$0.046\pm 0.019$	$0.194\pm 0.058$

Appendix

Table 8: Within-sample results on strength datasets (

s

m

, and

h

Panel of Figure 4 — Figure 4: Sensitivity of PCL-U to $\alpha$ , $\beta$ , and $\gamma$ on the synthetic 1-4-4-4, $s$ , and $h$ settings.

Hyperparameters	IHDP	Jobs	Twins	1/2/8-4-4-4	s	h
dim_in	$16$	$64$	$64$	$16$	$64$	$64$
$\alpha$	$0.01$	$0.01$	$0.01$	$0.05$	$0.01$	$0.01$
$\beta$	$10$	$1$	$100$	$1$	$100$	$100$
$\gamma$	$10$	$10$	$1$	$0.1$	$100$	$0.1$
scale	$0.05$	$0.05$	$0.05$	$0.05$	$0.3$	$0.3$
h_split	Yes	No	Yes	Yes	Yes	Yes

Appendix

Table 9: Hyperparameters.

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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 Causal Inference

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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

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