Organizations: Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong · Rational Intelligence Lab, CISPA Helmholtz Center for Information Security, Germany · Australian Institute for Machine Learning, Adelaide University, Australia · College of Computing & Data Science, Nanyang Technological University, Singapore
Instrumental variable (IV) and proximal causal learning (Proxy) methods are central frameworks for causal inference in the presence of unobserved confounding. Despite substantial methodological advances, existing approaches rarely provide reliable epistemic uncertainty (EU) quantification. We address this gap through a Deconditional Gaussian Process (DGP) framework for uncertainty-aware causal learning. Our formulation recovers popular kernel estimators as the posterior mean, ensuring predictive precision, while the posterior variance yields principled and well-calibrated EU. Moreover, the probabilistic structure enables systematic model selection via marginal log-likelihood optimization. Empirical results demonstrate strong predictive performance alongside informative EU quantification, evaluated via empirical coverage frequencies and decision-aware accuracy rejection curves. Together, our approach provides a unified, practical solution for causal inference under unobserved confounding with reliable uncertainty.
Figures & tables
Figure 1: Causal Graphs
Figure 2: Accuracy-rejection curve for log design (data size =200 ), with quantile =0.65,0.75,0.85
Appendix figures & tables21 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: ARC curves for the Demand design: tiny improvement from counting first stage uncertainty.
Figure 4: Demo of the first stage UQ in the sine design.
Figure 5: DAG of Proxy setting when the observed confounders exist.
Figure 6: Demo of synthetic data with data size =1000 . Left: log , middle: linear , right: sine .
Figure 7: Accuracy-rejection curve for sine design (data size =200 ), with quantile =0.65,0.75,0.85 .
Figure 8: Accuracy-rejection curve for linear design (data size =200 ), with quantile =0.65,0.75,0.85 .
Figure 9: Accuracy-rejection curve for demand design (data size =200 ), with quantile =0.65,0.75,0.85 .
Figure 10: Demo of demand data with data size =1000 . Left: the estimated f^(p,t=2,s=4) . Right: the estimated f^(p=20,t,s=4) .
Figure 11: Demo Plots of log design of Standard GP and GPIV with data size =200
Figure 12: MSE comparison of Standard GP and GPIV.
Figure 13: Active Learning of log (left) and linear (right) design.
lengthscale optimized
MSE
Coverage
AUC
(p,t,s)
.070 (.003)
.961 (.007)
.918 (.003)
(p,t,s,c)
.275 (.030)
.948 (.007)
.923 (.003)
(t,s)
.546 (.020)
.957 (.004)
.931 (.002)
(c,t,s)
.724 (.023)
.930 (.005)
.915 (.002)
∅
.642 (.033)
.851 (.005)
.895 (.002)
(c)
.906 (.043)
.842 (.006)
.866 (.003)
Appendix
Table 7: Effect of optimized lengthscales on MSE, Coverage, and Area under AUC with quantile =0.75 for demand design with 200 data.
Figure 14: Sensitivity analysis for various lengthscale of x for the log design with n=2000
Figure 15: Consequence for optimizing lengthscale of z for the log design with n=2000 : plots of MSE versus optimized lz .
Figure 16: Sensitivity analysis for various lengthscale of z for the log design with n=2000
Figure 17: Sensitivity Analysis on η
Figure 18: Plots of MSE and Coverage with 95% C.I. for θ from 0.00 to 0.55
Figure 19: Demo of Proximal synthetic (left) and demand (right) design with 1000 training data.
Figure 20: Accuracy-rejection curve for proxy synthetic design (data size =200,500,1000 ), with quantile =0.65,0.75,0.85 .
Figure 21: Accuracy-rejection curve for proxy demand design (data size =200,500,1000 ), with quantile =0.65,0.75,0.85 .
Design
n
Methods
sine
200
KIV
500
KIV, MMRIV
1000
KIV, MMRIV
log
200
KIV
500
KIV, QBIV
1000
KIV, MMRIV, QBIV
Appendix
Table 15: Methods with significantly worse performancen on MSE than GPIV (left) and GPProxy (right) under Wilcoxon signed-rank test, 5% .
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.
Ying Tang, Yi Wang
Shanghai University of International Business and Economics, Shanghai 201620, China
The presence of confounding bias poses a key challenge in policy evaluation, as the target causal effects of actions are not identifiable (i.e., underdetermined) from observational data. On the other hand, existing confounding-robust evaluation strategies require detailed prior knowledge about the environment or apply only to discrete treatments and outcomes. This paper investigates causal effect evaluation over the continuous domain from confounded observations, while requiring only basic temporal ordering between the treatment and the outcome. We introduce a universal discretization of the exogenous domains that approximates the observational and interventional distributions of any causal model with arbitrary accuracy using a finite number of latent states. Building on this newfound universal approximation property, we develop a novel family of Causal Gaussian process (CGP) models that effectively approximate the observational and interventional distributions of any causal model with confounded observations.
Junzhe Zhang, Jingyuan Chen, Elias Bareinboim
Department of Electrical Engineering and Computer Science, Syracuse University · Department of Computer Science, Columbia University
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.
Yonghan Jung
School of Information Sciences University of Illinois Urbana-Champaign