Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect

We thank the reviewer for recognizing the significance of our identification theory, influence function derivation, and optimal adjustment set analysis, as well as our motivation from fairness. We are grateful for the constructive suggestions that will help us further improve the completeness and clarity of our work.

Weakness 1: Simulation to validate efficiency gains.

We appreciate this valuable suggestion. In the revised version, we have conducted a series of simulation studies to evaluate the performance of AIPW estimators for the WCDE under finite sample settings. Details of the simulation setup and results are provided in our response to Reviewer tnMD (Additional Experimental Results). These simulations have been completed and will be incorporated into the revised version of the paper.

Our findings demonstrate that the optimal Valid Adjustment Set substantially reduces variance and improves estimation accuracy when the sample size is limited. In particular, the adjustment set $\{G_1, G_2\}$ consistently achieves the lowest variance and MSE across both backdoor and mediator adjustment scenarios.

Weakness 2: Discussion of DAG misspecification.

Thank you for highlighting this important limitation. We acknowledge that assuming a known DAG is a strong condition, and we will explicitly clarify this in the revised paper. As the reviewer notes, our framework is aligned with standard theoretical work in causal inference [1, 2, 3], where identification and efficiency results are derived under the assumption of a known graph structure. We will add discussion distinguishing this setting from causal discovery, which addresses DAG estimation under uncertainty [4, 5]. A more detailed analysis of this limitation and its practical implications is provided in our response to Reviewer Nm1A (Weakness 1).

Limitation: Real-world fairness case study.

We thank the reviewer for raising this important point. Indeed, both in theoretical analyses [6] and real-world applications [7], different mediator values $\mathbf{m}$ can lead to different CDE values, making interpretation challenging across subpopulations. In contrast, the WCDE is uniquely defined, as it averages over the marginal distribution of mediators rather than conditioning on a specific $\mathbf{m}$. This avoids ambiguity and provides a consistent estimand across populations. Additionally, by averaging over different mediator values, it has smaller finite sample variances.We will clarify this distinction and the practical advantages of WCDE in our revised paper.

[1] L. Henckel, E. Perković, and M. H. Maathuis. Graphical Criteria for Efficient Total Effect Estimation via Adjustment in Causal Linear Models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 84(2):579--599, 2022.

[2] J. Runge. Necessary and Sufficient Graphical Conditions for Optimal Adjustment Sets in Causal Graphical Models with Hidden Variables. arXiv preprint arXiv:2102.10324, 2021.

[3] A. Rotnitzky and E. Smucler. Efficient Adjustment Sets for Population Average Causal Treatment Effect Estimation in Graphical Models. Journal of Machine Learning Research, 21(188):1--86, 2020.

[4] J. Maasch, W. Pan, S. Gupta, V. Kuleshov, K. Gan, and F. Wang. Local Discovery by Partitioning: Polynomial-Time Causal Discovery Around Exposure--Outcome Pairs. In Proceedings of the 40th Conference on Uncertainty in Artificial Intelligence (UAI), 2024.

[5] J. Maasch, K. Gan, V. Chen, A. Orfanoudaki, N.-J. Akpinar, and F. Wang. Local Causal Discovery for Structural Evidence of Direct Discrimination. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 19349--19357, 2025.

[6] T. J. VanderWeele. Controlled Direct and Mediated Effects: Definition, Identification and Bounds. Scandinavian Journal of Statistics, 38(3):551--563, 2011.

[7] W. W. Loh, O. Lüdtke, A. Robitzsch, U. Trautwein, and O. Lüdtke. Estimation of Controlled Direct Effects in Longitudinal Mediation Analyses with Latent Variables in Randomized Studies. Multivariate Behavioral Research, 55(5):763--785, 2020.

We thank the reviewer for highlighting the clarity of the exposition and the paper’s relevance to fairness-related applications. We appreciate the reviewer’s thoughtful feedback and apologize for the aspects of the paper that were unclear or incomplete.

Weakness 1: Experimental results. We thank the reviewer for the excellent suggestions. In the revised version, we include new simulation results to empirically validate the efficiency of different valid adjustment sets in estimating the WCDE. Details of the simulation setup and results are provided in our response to Reviewer tnMD (Additional Experimental Results). The results consistently demonstrate that the adjustment set selected via our optimality criterion achieves lower variance and MSE compared to alternative valid sets, especially under finite samples.

Weakness 2: More illustrative figures.

We agree that additional visual aids would significantly enhance clarity. Condition 2.3(4) is indeed abstract, which is why we specifically included Example 2.5 and Figure 1 to illustrate its meaning. In the revised version, we will incorporate more causal diagrams to better explain Condition 2.3—highlighting the roles of the mediators, treatment, and outcome, as well as elucidating the purpose of the fourth condition.

Similarly, following Theorem 4.3, we will add a concrete DAG example to demonstrate how a sequence of variance-reducing operations leads to the construction of the optimal adjustment set (the O-set). This will help readers better understand the motivation behind the O-set and the roles of Lemmas 4.4 through 4.7. All figures will be clearly labeled and integrated into the main text for improved readability.

Weakness 3: Refinements in L133 and L145.

We apologize for the imprecision in the original version. The original notation "$\mathbf{M}' \subseteq \mathbf{M}$ be parents of $Y$" was indeed vague and could cause confusion. In the revision, we will clarify this by explicitly defining $\mathbf{M}' := \mathbf{M} \cap \text{Pa}(Y)$ to indicate that only the mediators that are parents of the outcome are included in the weighted contrast. This refinement ensures consistency with the identification logic and makes the derivation more precise.

Additionally, we will revise Line 145 to clearly state the counterfactual independence assumptions used in the derivation. Since WCDE is defined as a weighted version of the Controlled Direct Effect (CDE), it inherits the identification conditions from CDE. Let $Y_{a, \mathbf{m}}$ denote the potential outcome that would have been observed had the treatment been set to $A = a$ and the mediators set to $\mathbf{M} = \mathbf{m}$. Identifying the causal effect of $A$ on $Y$ while holding $\mathbf{M}$ fixed at $\mathbf{m}$ requires the following standard assumptions:

(1) No unmeasured confounding of the treatment–outcome relationship:

$$Y_{a, \\mathbf{m}} \\perp\\!\\!\\!\\perp A \\mid \\mathbf{Z}_2$$

(2) No unmeasured confounding of the mediator–outcome relationship:

$$Y_{a, \\mathbf{m}} \\perp\\!\\!\\!\\perp \\mathbf{M} \\mid A, \\mathbf{Z}_2$$

Here, $\mathbf{Z_2}$ represents a valid adjustment set for identifying the CDE. These assumptions ensure that, conditional on $\mathbf{Z_2}$, the joint intervention on $A$ and $\\mathbf{M}$ yields counterfactual outcomes that are independent of the actual treatment assignment and mediator values. This enables the identification of $\\mathbb{E}[Y_{a, \\mathbf{m}}]$ from observed data under standard causal models. We will incorporate these clarifications in the revised paper to enhance the precision and transparency of our derivation.

We thank the reviewer for recognizing the importance of WCDE and acknowledging the theoretical contributions of our work, and we appreciate the thoughtful feedback that highlights areas where further clarification is needed. We apologize for any aspects of the paper that were unclear or incomplete.

Weakness 1: Practical feasibility of determining the O-set. We thank the reviewer for raising this important point. Indeed, learning the causal DAG from data is a fundamental and ongoing challenge in causal discovery, often requiring exponential runtime or suffering from poor finite-sample performance [1, 2]. A more data-efficient alternative is to use local causal discovery algorithms that learn only the relative relationships between the exposure and outcome [3, 4]. For example, the algorithm proposed in [4] efficiently recovers an optimal O-set for WCDE without requiring full causal graph recovery under suitable assumptions. However, we note that these papers did not establish the asymptotic optimality of the proposed O-set, which is precisely the goal of this paper. As these papers demonstrate, developing reliable, data-efficient methods to determine the O-set is a nontrivial problem and beyond the scope of this work. Thanks to the reviewer's point, we will add the following discussion at the end of the paper:

In this work, we assume that the underlying DAG of the causal graph is known and accurate. While this assumption simplifies the identification of valid adjustment sets, we recognize that in practice, the DAG must often be learned from data—a challenge actively studied in the causal discovery literature. Existing global discovery algorithms can recover DAGs from observational data, but often suffer from poor finite-sample performance or exponential runtime in the worst case [1,2]. As a computationally efficient alternative, local discovery algorithms focus on identifying the relative relationships among exposure, and outcome, bypassing the need to learn the full global DAG [3, 4]. In particular, [4] propose a local discovery method tailored for WCDE estimation in fairness applications, under the assumption that the outcome variable is not causally downstream of any observed variables and that all of its direct causes are observed. They show that the proposed method has a polynomial runtime while allowing for unobserved confounders that do not directly affect the outcome. In this work, we establish that the set returned by [4] in their problem setting is indeed asymptotically optimal. Another promising approach to learn causal DAG from data is to assume an additive noise model (ANM), under which polynomial-time global discovery algorithms are provably consistent [5, 6, 7], potentially enabling efficient identification of adjustment sets.

While these alternatives offer practical avenues for discovering the O-set from data, a key open challenge is understanding how errors from causal discovery propagate to the final estimation of WCDE. Our work establishes the optimality of the O-set assuming a correct DAG, but integrating uncertainty from causal discovery into the efficiency analysis of adjustment strategies remains an important direction for future work. Developing robust frameworks that combine structure learning with optimal adjustment set selection would enable more principled and data-driven applications of WCDE estimation in practice.

Weakness 2: Other criteria beyond asymptotic variance.

We agree with the reviewer that criteria such as finite-sample bias, robustness, and computational efficiency are also important for practical applications. However, we recognize that analytical characterization of finite-sample properties is often intractable, especially under general nonlinear and high-dimensional settings. For this reason, we supplement our theoretical results with extensive simulation studies to empirically evaluate the variance, bias, and MSE of different valid adjustment sets across multiple DAGs and outcome mechanisms. Please see the detailed response to Reviewer tnMD (Additional Experimental Results). We will revise the paper to also include a discussion of related work that seeks to balance bias and variance in finite samples, such as Henckel et al.~[8], and recent empirical studies on adjustment set selection in finite samples. These additions aim to strengthen the connection between our asymptotic results and practical performance in finite-sample regimes, providing a more comprehensive view of optimal adjustment strategies.

Weakness 3/Question 1: Experimental Validation.

We thank the reviewer for this insightful comment. To address this point, we have conducted a series of simulation studies to evaluate the performance of AIPW estimators for the WCDE under finite sample settings. Details of the simulation setup and results are provided in our response to Reviewer tnMD (Additional Experimental Results). These simulations have been completed and will be incorporated into the revised version of the paper.

Our findings demonstrate that the optimal Valid Adjustment Set substantially reduces variance and improves estimation accuracy when the sample size is limited. In particular, the adjustment set $(G_1, G_2)$ consistently achieves the lowest variance and MSE across both backdoor and mediator adjustment scenarios.

Question 2: Robustness of results to uncertainty associated with DAG

In addition to our response to Weakness 1, we emphasize that identifiability and optimal adjustment sets depend on whether the estimated DAG and the true group DAG share the same optimal O-set — a quantity that is generally unverifiable in practice. A more practical approach to handling this uncertainty is to examine the resulting WCDE estimates through their bias and variance properties, though this ultimately relates back to the points addressed in Weakness 1.

Regarding the influence function, as shown in Theorem 3.4, while different DAGs may induce different valid adjustment sets $(\mathbf{Z}_1, \mathbf{Z}_2)$, the general form of the influence function for $T_a(\mathbf{Z})$ depends solely on the target parameter WCDE itself, rather than the specific DAG structure. We will emphasize this point in the revised paper to clarify the generality and robustness of our derivation.

Question 3: Use of the influence function in practice.

We thank the reviewer for this important question. Indeed, including an exemplary efficient RAL estimator for WCDE would significantly strengthen our paper.

In our experiments, we implemented the augmented inverse probability weighting (AIPW) estimator for WCDE. This estimator is constructed based on the influence function we derived in Theorem 3.4, with each component estimated using plug-in methods. Specifically, we estimated the relevant conditional means and densities, and combined them according to the structure of the influence function to form a doubly robust estimator. This ensures consistency if either the outcome model or the generalized propensity scores are correctly specified.

We will add a detailed description of this estimator to our revised paper. Moreover, we note that the influence function we provide also enables the construction of other efficient RAL estimators, such as TMLE or DML, using their respective estimation frameworks.

[1] Markus Kalisch and Peter Bühlmann. Estimating high-dimensional directed acyclic graphs with the PC-algorithm. Journal of Machine Learning Research, 8:613–636, 2007.

[2] Yang-Bo He and Zhi Geng. Active learning of causal networks with intervention experiments and optimal designs. Journal of Machine Learning Research, 9: 2523–2547, 2008.

[3] J. Maasch, W. Pan, S. Gupta, V. Kuleshov, K. Gan, and F. Wang. Local Discovery by Partitioning: Polynomial-Time Causal Discovery Around Exposure--Outcome Pairs. In Proceedings of the 40th Conference on Uncertainty in Artificial Intelligence (UAI), 2024.

[4] J. Maasch, K. Gan, V. Chen, A. Orfanoudaki, N.-J. Akpinar, and F. Wang. Local Causal Discovery for Structural Evidence of Direct Discrimination. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 19349--19357, 2025.

[5] S. Hiremath, P. Ghosal, and K. Gan. LoSAM: Local Search in Additive Noise Models with Unmeasured Confounders, a Top-Down Global Discovery Approach. In Proceedings of the 41st Conference on Uncertainty in Artificial Intelligence (UAI), 2025.

[6] S. Hiremath, J. Maash, M. Gao, P. Ghosal, and K. Gan. Hybrid Top-Down Global Causal Discovery with Local Search for Linear and Nonlinear Additive Noise Models. In Proceedings of the Neural Information Processing Systems (NeurIPS), 2024.

[7] Hoyer, Patrik, et al. "Nonlinear causal discovery with additive noise models." Advances in Neural Information Processing Systems, 21, 2008.

[8] L. Henckel, E. Perković, and M. H. Maathuis. Graphical Criteria for Efficient Total Effect Estimation via Adjustment in Causal Linear Models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 84(2):579--599, 2022.

We thank the reviewer for the positive feedback, particularly regarding the proof idea for identifying the optimal valid adjustment set and the provided four-step sketch. In what follows, we address all points raised.

Weakness 1/Question 1: Clarify "unique identifiability" and the existence of multiple adjustment sets.

We appreciate the reviewer’s suggestion. In the revision, we will clarify that "unique identifiability" means the WCDE is a well-defined causal quantity that yields the same value in the population regardless of which valid adjustment set is used (under Condition 2.3). Estimator variability arises because different adjustment sets can produce different finite-sample estimates of this quantity due to varying statistical properties (e.g., bias and variance). This distinction motivates our focus on selecting optimal adjustment sets to improve estimation quality while preserving the theoretical identifiability guarantee.

Weakness 2: Define $M$, and clarify the role of $A$ and $Y$ in Condition 2.3.

We apologize for the confusion. The definitions of $M$, $A$, and $Y$ are provided earlier in the subsection where Condition 2.3 appears (Lines 128–133). However, we agree that reiterating them within the condition itself would improve clarity. In the revision, we will explicitly define $M$ as the set of mediator variables in Condition 2.3. We will also restate that $A$ and $Y$ denote the treatment and outcome, respectively, and clarify that the condition is stated in terms of arbitrary disjoint sets to generalize beyond specific treatment-outcome pairs.

Weakness 3: Define "backdoor" and "mediator" paths.

We will include formal definitions of these concepts in the revised paper, referencing Pearl [1] to ensure consistency with the standard causal inference literature. Specifically, we define the following types of paths to clarify our assumptions:

– Directed path: A path in which all arrows point in the same direction, from the starting node to the end node. For example, $A \rightarrow B \rightarrow C \rightarrow Y$ is a directed path from $A$ to $Y$.

– Backdoor path: Any undirected path between $A$ and $Y$ that starts with an arrow into $A$. For instance, $A \leftarrow Z \rightarrow Y$ is a backdoor path from $A$ to $Y$ via $Z$.

– Mediator path: Any directed path from $A$ to $Y$ that passes through at least one mediator variable $M \in \mathbf{M}$. A typical example is $A \rightarrow M \rightarrow Y$.

These definitions will be formally added in Appendix A, along with illustrative examples to aid understanding. Additionally, we will correct the minor typo in the third item of Condition 2.3 (“are block”), and again we apologize for any confusion this may have caused.

Question 2: Discuss limitations.

We appreciate this valuable suggestion. In our revision, we will clarify that our method assumes the DAG is known and correctly specified. While in practice the DAG often needs to be estimated from data, we address the implications of this limitation more thoroughly in our response to Reviewer Nm1A (Weakness 1). We will include the study of our method’s robustness to DAG misspecification as an important direction for future research.

Additional Experimental Results

Further, to validate the robustness of our procedure under finite samples, we conducted additional experiments and will add them to our revised paper. We construct two 7-node DAGs with multiple causal pathways, corresponding to the scenarios in Lemmas 1–4 (Figures 3 and 4). Our data is generated as follows:

For each DAG edge, coefficients are sampled independently: with 50% probability from Uniform $[-1.5, -0.5]$, otherwise from Uniform $[0.5, 1.5]$, ensuring non-negligible effects. Each variable $X_i$ in the DAG is generated recursively according to a structural equation model:

$$X_i = \sum_{j \in \text{Pa}(i)} w_{ji} \cdot f_j(X_j) + \varepsilon_i,$$

where $w_{ji}$ is the sampled edge coefficient from parent $j$ to $i$. The function $f_j(\cdot)$ is applied independently to each parent variable $X_j$ before combination, and is selected uniformly at random from the following:

– Identity: $f(x) = x$
– Squared: $f(x) = x^2$
– Exponential: $f(x) = \exp(x)$

The additive noise term $\varepsilon_i$ is sampled independently from a Gaussian distribution $\mathcal{N}(0, 1/4)$ for each node with parents. Root nodes (i.e., nodes with no parents) are sampled from $\mathcal{N}(0, 1)$.

For each simulation run, we randomly select one of the three nonlinear functions $f$ to transform the outcome. The final outcome $Y$ is generated by applying the chosen $f$ to its parent variables. We repeat this process for 100 randomly generated coefficient sets, simulating each to assess the robustness of WCDE estimation.

Estimation Method. We implement an AIPW estimator with:

– Q-model: linear regression with spline-transformed features (degree 4, 20 knots)

– g-model: logistic regression for the treatment model.

The detailed form of the AIPW is included in the response to Reviewer Nm1A (Question 3).

Evaluation Metrics

For each adjustment set, we report average variance and average MSE across 100 replications. The first two tables correspond to Figure 3 and illustrate the results associated with Lemmas 4.4 and 4.6. The latter two tables correspond to Figure 4, which supports the theoretical findings in Lemmas 4.5 and 4.7.

Table 1. Average Variance for Figure 3

Table 2. Average MSE for Figure 3

Table 3. Average Variance for Figure 4

Table 4. Average MSE for Figure 4

Our proposed O-set $\{G_1, G_2\}$ consistently achieves the lowest variance and MSE across all sample sizes—from as small as $n=100$ to as large as $n=4000$—demonstrating its finite-sample robustness despite our theoretical guarantees being asymptotic. The current tables present preliminary simulation results, with more extensive experiments planned for the final version to strengthen empirical validation and provide a more complete set of results.

[1] J. Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, 2009.