Title: Coupling Generative Modeling and an Autoencoder with the Causal Bridge
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causal Effect Estimation

Proxy Causal Learning

Causal Bridge
Abstract: We consider inferring the causal effect of a treatment (intervention) on an outcome of interest in situations where there is potentially an unobserved confounder influencing both the treatment and the outcome. This is achievable by assuming access to two separate sets of control (proxy) measurements associated with treatment and outcomes, which are used to estimate treatment effects through a function termed the *causal bridge* (CB). We present a new theoretical perspective, associated assumptions for when estimating treatment effects with the CB is feasible, and a bound on the average error of the treatment effect when the CB assumptions are violated. From this new perspective, we then demonstrate how coupling the CB with an autoencoder architecture allows for the sharing of statistical strength between observed quantities (proxies, treatment, and outcomes), thus improving the quality of the CB estimates. Experiments on synthetic and real-world data demonstrate the effectiveness of the proposed approach relative to state-of-the-art methodology for causal inference with proxy measurements.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper derives theoretical bounds on the average error in treatment effect estimation under proximal causal inference when the causal bridge (CB) assumption is violated. The authors also propose an autoencoder-based architecture for estimating the causal bridge function, providing both theoretical guarantees and a practical implementation framework.\n\nSTRENGTHS AND WEAKNESSES:\n*Strengths*:\n- The paper extends existing theory for proximal causal inference by providing theoretical analysis for the scenario where the bridge function independence assumption (relative to the proxy variable) is violated.\n- The experimental evaluation includes both synthetic datasets and real-world data with ground truth established through randomized controlled trials, providing comprehensive empirical validation.\n\n*Weaknesses*:\n- The paper fails to discuss or compare against recent neural network approaches for unmeasured confounding, including key works such as Deep Learning Methods for Proximal Inference via Maximum Moment Restriction (Kompa et al., 2022), Causal Effect Estimation using Identifiable Variational AutoEncoder with Latent Confounders and Post-Treatment Variables (Xie et al., 2024), and Density Ratio-based Proxy Causal Learning Without Density Ratios (Bozkurt et al., 2025).\n- The synthetic data experiments omit important baselines: (1) CAVAE: the authors' claim that results are not numerically stable does not justify excluding this established method, and (2) the approach from Kompa et al. (2022).\n- The real-world data evaluation lacks comparison with relevant baselines, particularly Ying et al. (2022).\n- The authors present their method's proportional hazard assumptions as an advantage over Ying et al. (2022) in survival outcomes, but fail to acknowledge that requiring this assumption could be viewed as a limitation of their own approach.\n- The paper provides no justification for choosing autoencoders over diffusion models for sampling, despite diffusion models' growing prominence in generative modeling\n\nQUESTIONS:\nI might increase or decrease my score based on authors' responses\n- What motivated the choice of autoencoders over diffusion models for generating samples from the conditional distribution, particularly given diffusion models' demonstrated effectiveness in generative tasks?\n- Could you include comparisons with the missing baselines identified in the weakness section (CAVAE, Kompa et al. 2022, and Ying et al. 2022) to provide a more comprehensive experimental evaluation?\n\nLIMITATIONS:\nThe authors fail to acknowledge that their experimental evaluation omits several important baselines from recent literature, and do not discuss why they chose auto-encoders instead of diffusion models.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on the effect estimation with latent variables and two additional observed proxies,  a framework commonly referred to as proximal methods. The paper derives a new error bound on the causal bridge function (more specifically,  the outcome confounding bridge function), highlighting that the proxies should serve as low-noise measurements of the latent confounders. Further, the paper proposes a generative model based on an autoencoder architecture to learn the bridge function. Experimental results demonstrate the effectiveness of the proposed approach.\n\nSTRENGTHS AND WEAKNESSES:\n### strengths\n\n1. The paper is clearly written and well-organized.\n\n1. It proposes an autoencoder-based architecture for learning the causal bridge function within the proximal framework, which performs well empirically.\n\n   \n\n### Weaknesses / Questions \n\n1. Corollary is not a surprising result, since $\\Psi$ is a non-information-loss function. Therefore, the information loss stems solely from the exogenous noise $\\epsilon_w$.  As $\\sigma_{\\epsilon_w} \\rightarrow 0$, the proxy $W$ become equivalent to $U$, leading to $I(U;Z|W) \\approx I(\\Psi(W);Z|W)=I(W;Z|W)=0$. \n1. The authors discuss solely the case of outcome confounding bridge function $b$ (denoted $h$ in [1]). Is it possible to drive similar result for treatment  confounding bridge function $q$​ [1]? \n1. Relatedly, is it possible to learn the treatment bridge function $q$ using the proposed autoencoder architecture?\n1. It would be better to clarify why we should access $D_1$. What is the implication of using a separate dataset versus the same dataset?\n1. The experimental evaluation could be strengthened by comparing the proposed method with more state-of-the-art proximal baselines.\n1.  The paper discusses how the additional loss terms in Equation (13) improve the learned bridge function hhh, but an ablation study on these terms is missing.\n\n\n\n[1] Semiparametric proximal causal inference. JASA 2024.\n\nQUESTIONS:\nsee above\n\nLIMITATIONS:\nsee above", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents theoretical insights into the causal bridge method, a causal inference approach based on proxy variables, designed to estimate the causal effect of a treatment variable on an outcome in the presence of unobserved confounders. Building on these theoretical analyses, the paper proposes an architecture aimed at improving the quality of causal bridge estimates.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper is well-motivated and addresses an important problem. Causal effect estimation in the presence of unobserved confounding is a challenging yet critical issue, and proposing a framework to handle this is valuable.\n2. The paper is interesting, particularly the theoretical analysis and how it is used to justify the proposed method. The comparison with CEVAE is also compelling.\n3. The paper is well-structured and easy to follow and the proposed method demonstrates good performance, outperforming the baselines.\n\n\nWeaknesses:\n\n1. Some contributions of the paper appear incremental. In particular, much of Section 2, e.g., including Propositions 1 and 2, are direct derivations from Equation (1). Additionally, some findings, such as W and Z are noisy mappings of U have already been discussed in prior work [20].\n2. Some of the language in the paper needs to be more precise. For example, the introduction should explain more clearly the advantages of proxy variable methods, especially causal bridge, compared to others, such as instrumental variable approaches.\n3. The paper only uses one real-world dataset in the experiments. Including more real-world datasets would help better demonstrate the performance of the proposed method. Also, given the similarities between the proposed approach and the CEVAE, the paper should also include that as baseline. Also, I highly suggest that the paper includes more ablation studies in the main paper, rather than the appendix.\n\nQUESTIONS:\nPlease refer to Strengths And Weaknesses.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the challenge of estimating treatment effects under unobserved confounding by leveraging a causal bridge function within a proxy variable setting. It reformulates the bridge in a way that enables an intuitive deep learning implementation that improves the interpretability and statistical performance. The authors present theoretical conditions under which treatment effects can be identified using two proxy variables and introduce a novel set of assumptions that maintain the same inferential soundness. They further contribute a sensitivity analysis framework to assess robustness to assumption violations and demonstrate how their method enables statistical strength-sharing across variables. The framework consists of a two-step modeling pipeline: training a generative model to sample from proxy variables and then estimating treatment effects using these samples along with simulated unobserved confounders. A joint modeling strategy is employed to refine the confounder sampler. The approach is also extended to survival outcomes and is shown empirically to outperform state-of-the-art methods.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n1. The main contribution of the new bridge formulation is a novel result that cleverly allows a new practical implementation for estimating treatment effects.\n2. Assumptions are very clearly stated.\n3. Both the causal bridge approach and the joint objective are clearly demonstrated to lower estimate errors in out-of-sample settings in the experiments.\n\n**Weaknesses:**\n\n4. The paper is dense and is difficult to read. It would benefit from having figures and examples explaining the intuition of the bridge function, as well as the proven results.\n\nQUESTIONS:\nFull disclosure: I have previously reviewed this paper and gave it an accepting score. In the rebuttal, my questions were answered, and my opinion of the paper remains largely unchanged.\n\nLIMITATIONS:\nLimitations are stated.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}