Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect
Authors: Ruiyang Lin
Yongyi Guo
Kyra Gan
Venue: NeurIPS 2025 poster ()
Keywords: Optimal adjustment sets
causal inference
weighted controlled direct effect
nonparametric estimation
Primary area: probabilistic_methods

Abstract: The weighted controlled direct effect (WCDE) generalizes the standard controlled direct effect (CDE) by averaging over the mediator distribution, providing a robust estimate when treatment effects vary across mediator levels. This makes the WCDE especially relevant in fairness analysis, where it isolates the direct effect of an exposure on an outcome, independent of mediating pathways. In this work, we first establish necessary and sufficient conditions for the unique identifiability of the WCDE, clarifying when it diverges from the CDE. Next, we derive the efficient influence function for the WCDE and consider the class of regular and asymptotically linear estimators. We characterize the optimal covariate adjustment set that minimizes asymptotic variance, demonstrating how mediator-confounder interactions introduce distinct requirements compared to average treatment effect estimation. Our results offer a principled framework for efficient estimation of direct effects in complex causal systems, with practical applications in fairness and mediation analysis.

Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect
Ruiyang Lin The University of Science and Technology of China 1ruiyangorustl.en
Yongyi Guo
University of Wisconsin- Madison guo980wisc. edu
Kyra Gan
Cornell Tech, Cornell University
kyragan@cornell..em edu
Abstract
The weighted controlled direct effect (WCDE) generalizes the standard controlled direct effect (CDE) by averaging over the mediator distribution, providing a ro- bust estimate when treatment effects vary across mediator levels. This makes the WCDE especially relevant in fairness analysis. where it isolates the direct effect of an exposure on an outcome, independent of mediating pathways. This work establishes three fundamental advances for WCDE in observational studies: First, we establish necessary and sufficient conditions for the identifiability of the WCDE, clarifying when it diverges from the CDE. Next, consider nonparametric esti- mation the WCDE and derive its influence function, focusing on the class regular and asymptotically linear estimators. Lastly, we characterize the optimal covariate adjustment set that minimizes the asymptotic variance, demonstrating how mediator-come confoundee interactions introduce distinct requirements compared average treatment effect (ATE) estimation. Using synthetic and real-world data, validate theory numerically, showing that the proposed optimal valid adjust- ment set yields the lowest variance at practical sample sizes. Our results offer a principled framework for efficient estimation of direct effects in complex causal systems, with practical applications in fairness and mediation analysis.
Introduction
The controlled direct effect quantifies the effect of an exposure when intervening to set the mediator to fixed level, which may differ from its natural observed value [36, 39, 40J. While CDE has tradition- ally been used to analyze mediation effects, in this work, we employ the weighted controlled direct effect detect the presence of direct effect between treatment A and outcome Y given observed variables [31]. Specifically, we intervene on all observed mediators rather than an arbitrary subset. This approach is motivated by both pragmatic and theoretical considerations: e.g., from faimness perspective, we seek to determine whether direct discrimination exists [31], while methodologicall! the identifiability of CDE becomes unnecessarily complex when some mediators non-fixed [58], 2) CDE values can vary unreliably across mediator levels [53, 61], and 3) CDE estimates frequently vary across subpopulations defined by baseline covariates [23], complicating generalization.
Unlike natural direct effects (NDE) that require untestable cross-world independence assumptions (i.e., the potential outcome under treatment level is independent of the potential mediator under another treatment level) [1, 2, 41], the CDE, and by extension WCDE, relies only on experimentally verifiable interventions (where the mediator is set specific values through intervention) a crucial advantage noted by [38] and [57]. The WCDE averages CDEs over the mediator distribution, yielding
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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robust effect measure that automatically accounts for mediation interaction while maintaining the CDE's avoidance of cross-world assumptions. This provides critical insights into direct pathways across diverse domains, from evaluating medical treatments in epidemiology [11, [7] to assessing fairness [31, 62] and informing policy decisions [22].
WCDE raises two key statistical challenges in observational studies that should be resolved for valid inference: (I) What are the necessary and sufficient conditions on the adjustment set such that WCDE is uniquely identifiable from observed data? (2) Among all valid adjustment sets satisfying identifiability, which achieves the asymptotic efficiency bound for WCDE estimation? While optimal adjustment sets for ATES are rell-characterized [16, 34, 45], understanding WCDE introduces unique challenges due to the mediator's 's potential role as both a collider (between the exposure and the confounder) and an effect modifier (between the confounder and the outcome):
Standard CDE adjustment criteria [36] only ensure identification at fixed mediator levels and fail to address the integration required for WCDE (Example B.
ATE adjustment theory ignores the efficiency implications of mediator-confounder dependencies, which critically affect WCDE estimation variance through the weighting scheme.
This gap leads to potentially inefficient estimators and makes it unclear whether WCDE provides any theoretical advantages over conventional CDE in practice.
Contributions We study the identification and nonparametric estimation of WCDE given known directed acyclic gruph (DAG). Our contributions are fourfold. First, we establish necessary and suffi- cient conditions for valid adjusiment sets (VASs) that guarantee WCDE's identifiability (Lemma 2.6), a prerequisite for meaningful estimation. Second, we derive the inffuence function (IF) for WCDE (Theorem 3.4). Third, we prove that its optimal adjustment set-defined as the covariate collection that minimizes asymptotic variance among all regular and asymptotically linear (RAL) estimates- necessarily differs from ATE adjustment sets due to the required integration over mediator distribu- tions, and typically includes all parents of the outcome variable excluding the treatment (Theorem 4.3). These theoretical advances enable more precise direct effect estimation in settings with mediator- confounder dependence. Lastly, we provide an efficient estimator, augmented inverse probability weighting (AIPW), for WCDE, and numerically validate our theoretical results.
Outline Section 2 introduces notation, defines WCDE, and presents criteria for VASs. Section reviews nonparametric estimation and RAL estimators, and derives the IF of WCDE, The asymptotic variance of any RAL estimator is then determined by the variance of its IF, which depends explicitly on the choice of the VAS. Section 4 presents the optimal VAS for WCDE. Section 5 contains AIPW for WCDE and numerical results. Section 6 discusses connections to the unknown-graph setting.
Related Works Causal effect estimation from observational data typically relies on covariate ad- justment, with prior work examining variable selection through simulations [6, 28] and analyzing the minimum asymptotic variance [13, 43, 44]. Complete and sound graphical criteria for valid adjustment sets identifying total causal effects are well established [34, 37, 50), with later work optimizing asymptotic efficiency for ATFS under linear [16] and nonparametric [45] settings. Recent work has extended to individualized treatment rules and models with hidden variables [52]; establish- ing necessary and sufficient conditions for globally optimal adjustment sets [46]; and developing polynomial-timg algorithm for finding minimum-cost adjustment sets [51].
Beyond graphical criteria, data-driven methods for selecting sufficient adjustment sets include: 1) CovSel [8], which identifies minimal sets via conditional independence under correct model specification, 2) Outcome- Outcome-adaptive adaptive lasso [48], which selects variables predictive of the outcome to improve efficiency, 3) Double selection [4], which combines variables selected from both treatment and outcome models to ensure robustness in high dimensions. Estimator-driven approaches, such as Change-in-Fstine Entiiiatte [12] and Focused Confounder Selection [60], aim to minimize MSE but require parametric assumptions. These methods offer flexible and scalable alternatives when the causal structure is partially unknown, but their validity often hinges on model assumptions and sample size.
However, existing approaches remain centered on ATE and overlook key challenges posed by the WCDE. The mediator's dual role as a collider and effect modifier introduces unique obstacles to both identifiability and efficiency. While standard CDE-based criteria [59] do not guarantee identifiability of WCDE, ATE-oriented strategies treat mediators as forbidden variables [16, 45], precluding their integration as required by WCDE. We fill this gap by deriving graphical criteria and asymptotic efficiency results tailored to the nediator-weighteen structure of the WCDE.
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2 Problem Setup and Valid Adjustment Sets for WCDE
To formally define valid adjustment sets for the WCDE, we first introduce notation for graphical models. We then present the inferential framework and associated notation required to specify the estimation problem. Lastly, we classify variable types with respect to the exposure and outcome and provide a formal definition of VASs.
Structural Causal Model (SCM) An SCM is represented by DAG, 9 = (V, E), graph with directed edges and no directed cycles, where vertices V = (XI...., X2) represent random variables and edges E encode direct causal relationships. An SCM equipped with a probability distribution P over V. The parent set of vertex X,. denoted Pa(XX), consists of all vertices X, for which X; + X, E E, representing direct causes. The children Ch(X,) are vertices Xj with X; - X; E, representing direct effects. Ancestors An(X) comprise all vertices connected to X, via directed paths (Def. A.2), while descendants De (X;) are all vertices reachable from X, via directed paths.
The d-separation (Def. A.1) criterion formally characterizes conditional independence relationships implied by the graph structure. Under the assumption of causal sufficiency (no unmeasured confound- ing), d-separation perfectly captures the conditional independencies in the joint distribution through the Markov property: if Z d-separates X from Y in (i.e., X 1l Y Z), then X JL Y Z in all distributions that are Markov with respect to G. This Markov property implies two key consequences. First, the joint distribution factorizes as: P(X Xa) = I P(X; Pa(X,)) where each component represents the conditional distribution of a variable given its direct causes. Second, it yields the local Markov property that each variable is conditionally independent of its non-descendant non-parents given its parents:
X, TL ND(X)) \ Pa((X)) Pa(X)).
where ND( x3) denotes the non-descendants of Xj. These properties form the foundation for deriving identifiability conditions for causal effects in the presence of cendiator-coffoundee relationships.
Weighted Controlled Direct Effect Consider a DAG g = (V, E) representing a structural causal model. We focus on estimating WCDE of binary treatment variable A E V with values {a, a*} (treatment vs. control) on an outcome Y E V.
Let M C V be the set of all observed mediators between A and Y, defined through ancestral relations as M = De(A) n ( A, Y. The CDE measures the expected change in outcome as the exposure changes when mediators M are uniformly fixed to constant value through intervention:
Definition 2.1 (CDE, Pearl [35]). CDE(m) = EE(Y | do(a, m)] - E Y do (a", (a],))]
We provide the identifiability condition of CDE in Def. A.3. When mediator-exposure interactions present, CDE becomes rediaato--dppnnded taking different values across M [35]. To obtain an unique population-Ieve measure that admits valid adjustment, we define WCDE using the subset of mediators that are also direct parents of Y. ensuring that all mediator paths (Def. A.2) are blocked:
Definition 2.2 (WCDE, [31, 34)). Given a DAG G, let M' := M n Pa(Y) and let M' be the set of all possible joint values of the mediators in M.. The WCDE can be expressed using do do-probabilities as:
WCDE = Σ (L | do (a, m') EE [Y | do (a", m m'))) (m').
(2.1)
m EMM'
Definition 2.2 builds upon prior work by Pearl [34, 35), who established that fixing only the mediators that are parents of Y is sufficient for identifying direct effects.? 2 As M' is uniquely determined given G, for any given SCM, Eq. (2.1) admits an unique value assuming CDE's s identifiability.
Remark 2.3 (WCDE definition). Let C be the set of confounders between A and Y. In defining WCDE, we preserve the conceptual distinction between mediators and confounders, rather than treating mediators as additional confounders through joint conditioning. We weight the marginal distribution P(m') rather than the conditional P(m' (C) because the CDE(m') in Eq. (2.1) is already causally
IIn the first edition of Causality [34, Chapter 4.5.4, p. 131]. Pearl defined the average direct effect as Emy xx (B|Y do(x)), do(Pay \ X)] - EEY do(z'), do do(Pay \ X))) P(Pay \ X). Subscquent work [35] clarified that only mediators need to be fixed. Accordingly, we can replace Pay \ X with M' := M i Pa(Y). "In this work, we do not assume that all mediators are observed, Rather, a nonzero WCDE value indicates that direct effect between A and Y exists when accounting for all observed mediators.
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standardized over confounders C through the do-operator, representing the population-averas direct effect at mediator level m'. Weighting by P(m''C would inappropriately treat the mediator as confounder by reintroducing the mediator-confoumble creates conceptual redundancy by effectively conditioning twice on C. yielding stratun-specific effects conditional on C rather than a coherent population-Ievel measure.
relationship after causal standardization. This
The alternative factorization of the joint distribution, P(Clm'))Pnn) requires standardizing effects using P(C|m'), ), contradicting CDE's 's identification (Def. A.3). By using P(m'), we maintain philo- sophical clarity by respecting the distinct roles of confounders (adjusted for in causal standardization) and mediators (the pathway of interest). This provides a clear population-ley levell interpretation: the average direct effect under natural mediator variation.
While Def. 2.2 ensures WCDE identifiability, it does not specify how to compute it from observational data. We now characterize valid adjustment sets that recover the specific WCDE value in Eq. (2.1). While complete criteria exist for ATE adjustment sets [49, 50], no analogous solution currently exists for WCDE, gap we now resolve. We define VAS for WCDE in Def. 2.4, and provide a set of adjustment criteria in Condition 2.5. We establish their sufficiency and necessity in Lemma 2.6.
Definition 2.4 (Valid Adjustment Set for WCDE), Let Al be an SCM that induces causal DAG G with pairwise disjoint node sets A, Y, and M. representing the treatment, outcome, and the set of all mediators (Tab. A. I), respectively. Let Z₁ = n M, Z₂= = Z M, and
WCDEz = Ez, (Ez, E A = a, Zi Zell] Ez, E E AA a" Z1, Zell..
(2.2)
(Z)
T. (Z)
We say that Z is a valid adjustment set for identifying the WCDE with respect (A, Y) if and only if WCDEz equals the interventional WCDE defined in Def. 2.2 when both are applied to the population defined by M.
Condition 2.5 (WCDE Adjustment Criteria). Given the notation and SCM from Definition 2.4, candidate adjustment set Z satisfies the adjustment criterion relative to (A, for WCDE in G if:
CI All mediator paths (Def. A.2) for {A,Y) are blocked by Z,;
C2 The subset Z2 satisfies the sufficient and necessary graphical conditions for identifying the CDE with mediator Zn, which we surther discuss in Appendix A.I;
C3 (M') 3 g A, (Pa(Y) \ (M' U {A])} Zı, ensuring an unique weighting scheme across mediator levels (illustrated in Fig. B.3, Example B.
Lemma 2.6 (Sufficiency and Necessity of the Adjustment Criterion). Let Z be any set that satisfies Condition 2.5 for WCDE with respect to (A, Y) in G. Then Z is a VAS (Def. 2.4). Conversely, any adjustment set Z that identifies the WCDE in Eq. (2.1) must necessarily satisfy Condition 2.5.
Lemma 2.6 establishes that the WCDE, which is uniquely defined by Def. 2.2, can be identified through any VAS satisfying Condition 2.5, with the same population value regardless of which valid set is chosen. However, in finite samples, estimates can vary across sets due to differences in statistical properties like bias and variance. A proof is provided in Appendix I. with a sketch below.
Necessity. We focus on the necessity of C3, as C1 is inherent to the definition of WCDE, and C2 follows from CDE's identification. For given DAG G and adjustment set Z, suppose that Z satisfies C1 and C2, but violates C3, we show that there exists a joint distribution over the vertices V in g so that WCDEz defined in Eq. (2.2) does not match the WCDE in (2.1). Specifically, let W- = M' Z1, and C = Pa(Y) \ (M' U A). Because C3 is violated, W Kig A, Z, This conditional dependence results in discrepancy between (2.2) and (2. due bias term involving p(w 21) - w 21,a,c) c), which is non-zero whenever W Ki A,C Z, (see Example B.
Sufficiency. We show that if an adjustment set Z satisfies all three criteria in Condition 2.5, then Eq. (2.2) identifies the WCDE. We first decompose Z into mediators Z, and non-mediators Zz. and then replace Zz with C, which continues to identify the CDE and therefore preserves the expectation. Next add W = WMM) Z₁ to Z₁, which likewise leaves the expectation unchanged. After removing non-parent variables, the remaining expectations taken over C and M'. Substituting Z₂ C and Z, = M' into Eq. (2.2), and noting that C' identifies the CDE for the mediator set M', we obtain the WCDE expression in Eq. 12.
Given the identification of all VASs, a natural question arises: when multiple such sets are available, which one should we choose? In this work, we address this question by comparing VASs through

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the lens of asymptotic variance. Specifically, we characterize how the choice of VAS affects the asymptotic variance across a general class of estimators. Our objective to identify the optimal VAS that minimizes asymptotic variance for any estimator within this class.
3 RAL Estimators and Influence Function of WCDE
To identify the optimal VAS, we begin by introducing regular and asymptotically linear estimators class of estimators with well-characterized asymptotic variance through their influence functions. These estimators provide the foundation for our optimality analysis, which we develop fully in Section 4 through constructive intaanceminimmiztion procedure. We then specialize this framework to WCDE estimation, presenting concrete examples of RAL estimators. Theorem 3.4 establishes the explicit form of the WCDE influence function as function of the chosen VAS.
RAL Estimators We begin by introducing RAL estimators in a general statistical framework. Given i.i.d. observations O1....., On, drawn from an unknown distribution P on sample space Ω. Our goal is to estimate real-valued target parameter, T(P). based on these observations. For example, in WCDE, for given valid adjustment set Z, the target parameter is given by Eq. (2.2). We consider a fully nonparametric model class M, i.c., P E MM, which contains all candidate distributions defined on measurable space (S2, F) dominated by common o-finite measure Let p denote the corresponding density of P with respect to V.
We first introduce the notion of regularity. which connects estimators to their stability under small model perturbations (Def. 3.1). Next, we define asymptotic linearity (Def. 3.2).
Definition 3.1 (Regular estimator, van der Vaart [56]). Let T(P): M - Ie be a target parameter
of interest. An estimator T, is said to be regular in M at P if its convergence to T(P) locally uniform in neighborhood of P in M.
Definition 3.2 (Asymptotically linear estimator, van der Vaart [56]). Given OI,..., On ltd. P and target parameter TP)), an estimator P, Pr.(O...., On) TP) is said to be asymptotically linear at P if there exists a function up E L)(P) such that vn(T,, - T(P)) tt Ei (O:) (1), where LP(P) is the set of all mean-zero, square- qquare-integrable functions funtiooss under P, i.e, LQ(P) = {f: EP(f(O) < co, Erlf(O)) =03 =0]
Common RAL estimators include the ordinary least squares estimator [33], inverse probability
weighting [42], AIPW estimator [10], doubly robust estimators [3, 9], targeted maximum likelihood
estimator [55], and double machine learning [7].
Influence Function The function thp in Definition 3.2 above is known as the influence function (IF)
of Tn at P. By the Central Limit Theorem, we have:
Vn - (TP)) $ N (0, Varp [p(())).
(3.1)
In fully nonparametric models, this IF is unique for all RAL estimators and coincides with the influence function of the target parameter T(P). At high level, IF of the target parameter T(P) measures the sensitivity of T (P) under small perturbations of the distribution P [54]. We provide formal definition in Appendix B.3.
Remark 3.3 (Influence Functions of WCDEz). We emphasize that Eq. (2.2) defines distinct target
parameters for each adjustment set Z. While these parameters coincide in value at the true distribution P (yielding an unique causal estimand), different choices of the adjustment set Z would affect the sensitivity of the corresponding target parameter WCDEz to local distributional perturbations (see concrete definitions of local distributional perturbation in Appendix B.3). Intuitively WCDEz is sensitive to perturbations in variables within Z. so the choice of Z directly affects its sensitivity to different perturbations. This leads to distinct influence functions for WCDEz across different Z. We formalize this argument in Appendix B.3.
Application to WCDE Given valid adjustment set Z, let Z, Zn first provide the IF corresponding to the first component of WCDE defined in Eq. (2.2), denoted by T. (Z)
n M and Z₂ = Z M. We
The IF of T.. (Z) can be similarly obtained by replacing with a² in the expression.
Here, denote O generic random variable drawn from the distribution P.
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X1
Group
Description
X5
X4
X
Confounders and their proxies.
X2
Colliders and their proxies. Mediators and their proxies.
X3
X4
Non-descendants of Y where X, Il and X A. A Y.
X
X2
X6
X5
Instruments and their proxies.
Xs
X
Descendants of Y with A-Y paths me- diated by Y.
Xa
The partition visualization follows Maasch et al. [32], with directed squiggly edges indicating the presence a directed causal path (not necessarily direct parent- child relationships). The dashed edge rep- resents potential parent-child relationships.
Figure 1:
X,
Descendants of A.
Xs
Nodes with no active paths A or Y.
Table 1: Exhaustive and Mutually Exclusive Causal Partitions [32]. Detailed definitions are included in Appendix A.2.
Theorem 3.4 (IF of WCDEz). Given any VAS Z, the influence function TZ given by Ta( Ia(A)p Zn (1)p(())))))))))))))() (Zz)
(y Ey[y|A = =a, Zz, Zz]
Va (Y, Zz; P)
Z1. Zz. a)
+ Ezz
Ey [Y a, Zi. Zz] Ez, Ey [Y|A= a. Z1 Zz]
2T, (Z)(Z)
Here, I(-} denotes the indicator function, and (Z) evaluated at
The proof of Theorem 3.4 begins by substituting this functional To (Z) into the definition of pathwise differentiability (see Appendix B.4 for details). Leveraging the fact that the influence function is unique in a fully nonparametrie model, we directly obtain an unique influence function associated with each valid adjustment set.
4 Optimal Valid Adjustment Sets for WCDE
Given that distinct valid adjustment sets typically yield different IFs (and thus different asymptotic variances), we now leverage the adjustment criterion from Section 2 and the IF derived in Section 3 identify the optimal VAS-the one minimizing asymptotic variance for all RAL estimators.
To facilitate the presentation our results, we adopt the causal Given causal DAG G, let (A, S V be exposure-outcome pair, where Y is nondescendant of A. This local taxonomy decomposes the rest variables the DAG, X = V \ {A, Y), into eight exhaustive and mutually exclusive subsets, each defined by the types of causal paths that can share with A and Y. When the union of multiple partitions is needed, we write, for example, X1,3 X, U X3. Moreover, for any & (1,...,8 8), denote by X XksPa(Y) Xx Pa(Y)) the subset of X that also belongs to the set of parents of Y.
causal-partition partition taxonomy in Table 1 [32].
Under the causal partition taxonomy, construct the set O as the union of all parents of Y belonging to relevant partition groups in Definition 4.1.
Definition 4.1 (O-set), Lei O(A,Y, Y,G) denote the set of all non-treatment purents of Y in G. For- mally, under the causal partition taxonomy, these parents fall into three groups: confounders (X1). mediators (Xa), and informative non-descendonts (Xa):
O(A,Y,)) = X1er((Y) U USsero(Y) U XXEPO(Y)-
Intuitively, ((A,Y9) constructed to maximize information on Y, while minimizing information on A and preserving validity. Next, we formally define the notion of optimal VAS in Definition 4.2. Theorem 4.3 establishes the optimality OAA,Y,)).
Definition 4.2 (Optimal VAS). Given DAG G (V, E) and an exposure outcome pair (A, Y) C
V, let Z denote the collection of all VASs identifying the WCDE of A on Y. We say that O E is an optimal valid adjustment set if. for every Z E Z, ( -8. (P) 2 0, where o (P) denotes the asymptotic variance of any RAL estimator of WCDEz based on adjustment set Z under distribution P. as characterized by Equation (3.1).

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B2
G2
C2
A
Figure 2: A DAG illustrating Lemmas 4.4, 4.6.
Figure 3: DAG illustrating Lemmas 4.5, 4.7.
Theorem 4.3 (O-set is Optimal), Given a DAG G = = (V, and the exposure-outcome outcome pair (A,Y) S V, the set ( P(A,Y,9 9) in Definition 4.1 is the optimal valid adjustment set for the WCDE (2.1), satisfying Condition 2.5 and producing minimal asymptotic variance for all RAL estimators.
Below, we sketch the main proof idea (full proof in Appendix C). Starting from an arbitrary VAS Z with Z1 ZnM and Z- Z M, iteratively transform it into (A, Y, g) through four variance- reducing operations: (1) adding non non-mediating parents of Y, (2) including mediating parents of Y, (3) removing redundant backdoor variables that do not contribute to Y, and (4) pruning mediators that offer no further gain. Each step preserves the validity of the adjustment set and reduces the asymptotic variance, ultimately converging to the same optimal set O regardless the starting VAS. While this sequence of variance-reducing operations resembles the pruning procedure proposed by [16] and the graphical criteria introduced by [45], our work extends these ideas to a new functional that involves mediator-confounder interactions, introducing additional structural complexity.
Step Augmenting Ze with Non-mediating Parents of Y. Lemma 4.4 (proof Appendix C.I) establishes that including additional backdoor variables satisfying conditional independence with the treatment can reduce variance without losing validity, i.e., if we add XIePa(Y) and XAEPa(Y) Zz:
o.., X (EENt)))) S (z))
Lemma 4.4 (Supplement of backdoor variables). Given DAG 9 (V,E), and the exposure- outcome pair (A, Y) C V, suppose (G1, B2) CVV (A,Y) is VAS, with G2 disjoint from both G1 and B2, satisfying G2 l.. A, G, B2. Then (G1, G2, B2) is also VAS, and for all P M,
of (G..B.) -G (G, B2) (P) 2.0
Proof Sketch of Lemma 4.4. Figure 2 illustrates the conditions stated in Lemma 4.4. For clarity, we focus on the first component of the WCDE Eq. (2.2), and let ( denote its asymptotic variance. The extension to the full WCDE is direct. We begin by decomposing the variance:
2 B.) (P) Varp [1%a (Y, A, G1, B2; P)] Varp [ba (Y, A, G1, G2, B2; P)]
Varp (V, A, G1, B2; P) (Y, A, G1, G2, B2; P)]
+2 Covp (Y, A, G1, G2, By; P), (Y, (Y,A, A, G1, By; P) V. A, G1, G.2, B)) To analyze the covariance term, consider the difference in IFs evaluated (Y, A, G1, B2):
a A, G1. B2: P) (Y, A, G1, B2;
We construct parametric submodel that perturbs only the conditional law P(A, G3 while leaving all other components unchanged. Under the assumption G2 و7 A, G. above difference is orthogonal to the IF of the larger adjustment set:
G2)
B2, the
Er ( G1, G2, B2; (Y, A, G2 B2; P) (Y,Y, G1, G2, B2; P))] = 0.
Hence, the covariance term vanishes. Substituting this result back into the law of total variance yields:
Variya (Y, G1, B2; P)] Varlu, A, G1, G2, P)]
+ "Yr(,,, G1, B2; vu A, G1, G2, B2; P),
where the second term nonnegative, This establishes "".,G,.B,, (P) 2 a² G.. (P), showing that including G2 either improves or preserves estimation efficiency.
Step 2: Augmenting Z with Mediating Parents of Y. Next, Lemma 4.5 (proof in Appendix C.3) states that including mediators that are also parents of Y leads improved efficiency. Therefore, we augment Z, with Xagra(Y):
o (2 ZzUXiemmy (xeen))))))))))) oz Z2UX (X
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Lemma 4.5 (Supplement of mediator variables), Given a DAG 9 = (V.E), and the exposure- outcome pair (A, C V, suppose (B,, G2) C VV a VAS, with G disjoint from both B, and G2, salisfying G 1l A, G2 B₁. Then (G1, B1, G2) is also VAS, and for all P E M, o (B....) GG G2) (P) 220.
Figure 3 provides a simplified illustration of the conditions stated in Lemma 4.5, The proof proceeds analogously to that of Lemma 4.4, with G2 playing the same role as Gg.
Step 3: Pruning Redundant Backdoor Variables from Zz. Lemma 4.6 (proof in Appendix C.2) implies that unnecessary variables in Z2 (those not in (iePa(Y) XaePa(Y)) can be removed without increasing variance:
(Z, ( XieBor, U (P) S o (Z,UXSERM)) X UX (P))
Lemma 4.6 (Deletion of overadjustment backdoor variables). Given DAG G = (V, E), and the exposure-outcomee pair (A S V. and suppose (G1, G2 U2) C VVA Y a VAS, with G and B disjoint, and suppose Y وا B2 G1. G2, A. Then (G,, G2) is also a VAS, and for all P E M, o (P))
&e G.). (P) 20 0.
Proof Sketch of Lemma 4.6. Figure 2 provides simplified illustration of the conditions stated in Lemma 4.6. Again, we focus on comparing the asymptotic variance of Ta (Z). We begin by applying the law of total variance to decompose the variance under the larger adjustment set (G1, G2. B2):
Varlva (Y, A, G1, G2, B2; P)] = Var (Y, A, G1, G2, B2; P) J,Y, G1, G2|] + E [ar[(va (Y, A, G1, G2, B2; P) A, G, G₁, G2]]
Now, under the conditional independence assumption Y Lg B2 G.. G2, A, the IF based on the adjustment set (G1, G2, B2) satisfies (v (Y, A, G1, B2, P) A,Y, G1, G2] = Va (Y, A, G2; P). Substituting this into the variance decomposition yields:
Varlv (Y, A, G1, G2, B2; P)]
= = (aalaaaY A, G1, G2; P)] + E [Var[ea AY,
A,Y, Gz]]
Since the conditional variance term is nonnegative, conclude that o (P) > o2 G.. implying that B2 can be safely excluded without increasing the asymptotic variance.
Step 4: Pruning Redundant Mediating Variables from Z, Finally, Lemma 4.7 (proof in Appendix C.4) shows that variables in Z₁ X3EPa(Y) can be removed:
( Xierary UX xemoy)) (P) oz FiZ,,XserYV x UX (P).
Therefore, the final adjustment set minimizing the asymptotic variance is given by
O XiePa(Y) U 3X3EPa(Y) U (1IEP(YY)
Lemma 4.7 (Deletion of overadjustment mediator variables). Given a DAG -V,G= E), and the exposure-outcomee pair (A, Y) C V, suppose (G, U B,, G2) C VVA,Y is VAS with G and B disjoint, and suppose g B, G1, G2, A. Then (G1, G2) also VAS, and for all P E M. 2
2 (.G. - G.G...)) (P) 1 0.
Figure provides a simplified illustration of the conditions stated in Lemma 4.7. The proof follows the same argument as Lemma 4.6, with B playing the same role as Bg
5 Experiments
To evaluate the robustness of our method under finite- sample conditions, we conduct synthetic experiments to verify Lemmas 4.4-4.7, corresponding to Figures and 3, and further demonstrate the practical stability of the O-set using real-world data.*
*Code available htttps//github.co. Ruiyang/WCDD. Swm.tmittteiiiimmmmmmmmmmmma

---

AIPW Estimator In our experiments, we implemented the AIPW estimator, constructed using the plug-in components of the influence function:
1 2n
IfA: = a} p ((z).) ((Ze))
WCDE
(Y, ja Zie. Z2))
Σ M(a, Zi.. z2) f(z2)
PZZii Zzi a)
IPW term at level
Marginal over Ze
=A,=aa) -0(Z))()))))))))))))))))) p(Zzs)
+ Σ Maa 21, Zoz)) p(zz)
(Y, - Maaa Zi.. Zo,))
2n
PiZni, Zei,
21
IPW term level
Marginal over Z, 2t
iia" Zii, 22) f((2)) +
Σ ii(a", 21, Z2i) f((1)
(5.1)
Marginal rver Zg at
Marginal over Zg at
To estimate the conditional expectations jaaa, Z, Z2), we fit linear regressions with spline- transformed features (degree 5, 10 knots) for continuous outcomes, and use multinomial logistic regression for discrete outcomes. The treatment probabilities p(A = a Z1. Zz) are obtained from logistic regressions. The marginal and joint densities are estimated nonparametrically, using empirical frequencies for discrete variables and Gaussian kernel density estimation for continuous ones.
Synthetic Experiments Under Figures 2 and 3, data are generated from a nonlinear structural equation model with random edge coefficients, nonlinear transformations, and additive Gaussian noise. Appendix D.1 contains full details of the data- Mata-generating process and estimator implementation. For each adjustment set, we report average variance and average MSE across 50 replications. Table top and bottom respectively illustrate the results associated with Lemmas 4.4 and 4.6, and Lemmas 4.5 and 4.7. Our proposed O-set {G1, G2 consistently achieves the lowest variance and MSE across all sample sizes, ranging from = 250 to = 2000, demonstrating its finite- nampe robustness despite our theoretical guarantees being asymptotic.
Table 2: Average Variance and MSE for Figures 2 and 3. Optimal VAS are highlighted in bold.
n
250
n 500
n
1000
n
2000
Figure Adjust. Set
Var
MSE
Var
MSE
Var
MSE
Var
MSE
G1, G2
0.0029
0.0030
0.0016
0.0016
0.0009
0.0009
0.0006
0.0006
Fig
G1, B2
0.0102
0.0103
0.01
0.01
0.0101
0.0102
0.0075
0.0076
G1, G2, B2
0.0050
0.0051
0.0028
0.0029
0.0021
0.0022
0.0016
0.0016
G₁ G2
0.0027
0.0028
0.0015
0.0015
0.0010
0.0010
0.0006
0.0006
Fig 3
B1, G2
0.0100
0.0101
0.0072
0.0073
0.0066
0.0067
0.0050
0.0050
G1, NT,BI.G, G2
0.0059
0.0059
0.0040
0.0041
0.0040
0.0040
0.0029
0.0029
Table 3: Variance and MSE of WCDE estimates on the ASIA network under different sample sizes.
Variable names are abbreviated as follows: bronc, s: smoke, 1: lung.
250
n
500
1000
n
4000
10000
Adj. Set
Var
MSE
Var
MSE
Var
MSE
Var
MSE
Var
MSE
[b]
0.0143
0.0158
0.0049
0.0052
0.00266
0.00292
0.00062
0.00067
0.00027
0.00029
[s]
0.0210
0.0238
0.00862
0.00911
0.00408
0.00454
0.00101
0.00104
0.00046
0.00054
[b.s]
s]
0.0176
0.0229
0.00628
0.00728
0.00419
0.00480
0.00087
0.00091
0.00044
0.00054
L.
0,0296
0.0528
0.0174
0.0231
0.0141
0.0153
0.00333
0.00335
0.00177
0.00194
[b. 0.0360
0.0540
0.0207
0.0248
0.0133
0.0139
0.00340
0.00342
0.00170
0.00183
[]] 0.0532
0.0624
0.0205
0.0215
0.0133
0.0137
0.00346
0.00344
0.00184
0.00193
V. 8]
0.0458
0.0613
0.0218
0.0240
0.0151
0.0157
0.00367
0.00368
0.00189
0.00204
Real-World Experiments We evaluate our method on three widely used semi-synthetic Bayesian networks from bnlearn: ASIA, SIGNALING, and MILDEW, which serve as standard benchmarks for causal inference due to their realistic structures and domain relevance [24, 27, 47]. For each dataset, we validated our theoretical results through five-step process: (1) selecting a treatment and outcome variable (for multi-valued treatments, the two most frequent levels are used to define binary treatment); (2) identifying all valid adjustment sets for small/medium DAGs and a sufficiently large and representative subset for large DAGs; (3) estimating the WCDE via the AIPW estimator

---

(Eq. (5.1)); (4) simulating data from the underlying causal model across sample sizes 250, 500, 1000, 4000, and 10000; and (5) computing the empirical variance and MSE for each adjustment set.
The empirical validation across three distinct networks consistently affirms the efficacy of our identification criterion. In the ASIA network (an 8-node 8-edge Bayesian network for medical diagnosis), our criterion identified bronc as the optimal set (O-set) for estimating the effect of single composite variable-iidicattini of breath). As shown in Table 3, brone attains the lowest variance and MSE across all sample sizes, consistent with our asymptotic theory and demonstrating robust
the presence of either tuberculosis or lung cancer-on dysp (shortness
performance. In the SIGNALING network (a 1- --ndee 17-edge DAG on protein signaling cascades), our criterion identifies (PKA, Erk) as the O-set for estimating the effect of PKC on Akt. Table illustrates a nuanced Slas-variance trade-off. For larger sample sizes (n 2 1000), our O-set achieved the smallest variance among all adjustment sets, though its MSE was not the absolute lowest. When the sample size was 500 or smaller, the O-set, while not achieving the very lowest variance or MSE, delivered performance that was highly comparable to the pest-performing sets, demonstrating its practical utility and robustness even in finite-sample regimes where strict optimality is not guaranteed. Similar to the ASIA network, in MILDEW (a 35-node 46-edge DAG on crop disease management), the identified O-set ([meldug_3 3, midde_ 3, mikro __3 for estimating the effect of mikro_ I on meldug 4) achieves the smallest variance and MSE when n 2 500, and remains among the best rest-performing sets for smaller samples (Tables D.2 and D.3 in Appendix D.2).
Table 4:
Variance and MSE of WCDE estimates on the SIGNALING network under different sample sizes, Variable names are abbreviated as follows: e: Erk, P: PKA, m: Mek, r: Raf.
250
500
1000
4000
n
10000
Adj. Set
Var
MSE
Var
MSE
Var
MSE
Var
MSE
Vur
MSE
[e. p]
0.00491
0.00618
0.00269
0.00328
0.00125
0,00134
0.00033
0.00036
0.00012
0.00012
Ie, p.
0.00483
0.00569
0.00293
0.00344
0.00156
0.00164
0.00041
0.00044
0.00019
0.00019
[e, m.pl
0,00429
0,00476
0,00238
0,002600
0.00142
0.00146
0.00062
0.00064
0.00020
0.00020
[c, m. p. r]
0.00456
0.00490
0.00272
0.00289
0.00159
0.00160
0.00065
0.00067
0.00020
0.00020
Im, p. rl
0.00843
0.01024
0.00486
0.00586
0.00383
0.00407
0.00205
0.00217
0.00105
0.00105
[m.p]
0.00752
0.00941
0,00408
0.00534
0,00351
0.00386
0.00201
0.00213
0.00108
0.00108
6
Discussion
We provide the first comprehensive framework for identifying and estimating the WCDE in obser- vational settings, where all observed mediators are fixed. We leave the analysis of more complex WCDE parameters, where only subset of mediators is fixed, to future work. WCDE introduces unique challenges stemming from the integration over mediator distributions and the dual role of mediators as both colliders and effect modifiers. To address these challenges, we first establish graphical criterion that characterizes VASs for WCDE. Second, we derive the unique IF of WCDE under nonparametric models, enabling principled comparison of valid adjustment strategies. Third, we identify the optimal VAS that minimizes asymptotic variance, showing that it systematically differs from those used for ATE estimation due mediator-confounder interactions. Together, our identification results and efficiency analysis lay the foundation for extending WCDE estimation to more complex settings where careful handling of mediator confouuner structure is essential.
While we assume that the underlying DAG is known and accurate, in practice, the DAG must often be learned from data, a challenge actively studied in the causal discovery literature. Global discovery algorithms can, in principle, recover DAGS from observational data, but they often suffer from poor finite-sample performance or exponential runtime in the worst case [15, 25). As a computationally efficient alternative, local discovery algorithms focus on the relative relationships between exposure and outcome, bypassing the need to recover the full DAG [31, 32]. Maasch al. [31] proposes a local discovery method tailored for WCDE estimation under suitable assumptions, showing polynomial runtime while allowing for unobserved confounders that do not directly affect the outcome. Our results further establish that the set returned by their procedure is asymptotically optimal in their problem setting. Another promising approach is to assume an additive noise model, under which global discovery algorithms are provably consistent in polynomial time [18-20] potentially enabling the efficient identification of adjustment sets. While these alternatives offer practical avenues for learning the O-set from data, key open challenge is to understand how errors from causal discovery propagate into WCDE estimation. Developing robust frameworks that integrate structure learning with optimal adjustment set selection remains important direction for future work.
10

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Acknowledgments
The authors wish to thank Ezequiel Smucler and Stephan Poppe for insightful discussions that helped improve this work. Y.G. is supported by NSF DMS-2515285. This work was supported in part by an AWS Cloud Credit Grant from Cornell's Center for Data Science for Enterprise and Society.
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Appendix
Table of contents
A Definition, Notation and Graph Terminology
22
A.I Sufficient and Necessary Conditions for CDE identification C2
23
A.1.1 Necessary and Sufficient Adjustment Criteria for Identifying do coo-robbbiiityy
23
A.1.2 CDE Adjustment Criteria
24
A.2 Definitions of Causal Partition Introduced by Maasch al. [32]
25
Proofs and Extended Discussions for Sections 2 and
26
B.1 Proof of Lemma 2.6
26
B.2 Illustration of Criterion C3
28
B.3 Discussions on Influence Functions
29
B.4 Proof of Theorem 3.4
31
Proofs of Results in Section
34
C.1 Proof of Lemma 4.4
34
C.2 Proof of Lemma 4.6
41
C33 Proof of Lemma 4.5
43
C,4 Proof of Lemma 4.7
49
C.5 Proof of Theorem 4.3
52
Experimental Details
53
D.1 Synthetic Experiments Data-Generating Process
53
D.2 MILDEW Network
53
A Definition, Notation and Graph Terminology
Definition A.1 (d-separation). For disjoint vertex sets X, Y, and Z, we say Z d-separates X from Y
every path between X and Y is blocked by Z. A path blocked by Z if it contains either:
(1) chain X; + Z% + Xj or fork X; x- Zu + X; with Zu E Z, or
(2) A collider X; - Z + X3 where Zk Z and none of its descendants are in Z, i.e.
De(Zx) nz 0.
Definition A.2 (Path types). To ensure consistency with the standard causal inference literature [34, 371, we define seven types of paths in a directed acyelic graph (DAG) G = =V,E) connecting treatment A and outcome Y:
(1) Directed path: path which all arrows point in the same direction, from the starting node to the end node. Example: A + B - C Y.
(2) Backdoor path: Any undirected path between and that starts with an arrow pointing into A. Example: A + 4 -Y
(3) Mediator path: Any directed path from Y that passes through at least one mediator variable M E M. Example: A M + +Y
(4) Causal path: path from to Y which all arrows are oriented away from A and toward Y.
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(5) Proper causal path and causal nodes: A causal path is proper (with respect to A) if only its first node lies in The set of all nodes lying on proper causal paths from A to Y, excluding A üself, is called the set of causal nodes and denoted by en( in(.Y.9).
(6) Non-causal path: Any path between and Y that is not causal, i.e., that starts with an arrow pointing inlo A.
Intuitively, directed and causal paths transmit causal influence from A to Y, whereas backdoor and other non-causal paths represent spurious associations that must be blocked for unbiased causal estimation.
Definition A.3 (Identification of the controlled direct effect), Following the notation of Vander Weele and Vansteelandi [59], let (a, m) denote the potential outcome that would have been observed had the trealment been sel to A = a and the mediator to M = m.. Let C denote (possibly multivariate) set of pre-exposure covariates. Identification of the CDE relies on the following counterfactual independence assumptions:
(1) No unmeasured confounding of the treatment cottome relationship:
((a LL Cl,
(2) No unmeasured confounding of the mediator- relationship:
Y(a, m) TL M A,C.
Under these assumptions, the controlled direct effect of changing from to a² while holding M fixed at m is identified as
CDE(a, mm;
[E (a, m) Y(aa, ,)]]
/ E|Y A = a, M = m, C - E A= a", M = m. C = cl) dFc(c).
(A.1)
A.1 Sufficient and Necessary Conditions for CDE identification C2
To characterize the identification of the controlled direct effect (CDE), we first revisit the general conditions under which post-intervention density involving the do-operator can be expressed function of observable conditional densities. These general graphical concepts-includdig adjustment sets, forbidden sets, and the adjustment criterion provide the foundational framework for identifying causal effects such as the CDE.
A.1.1 Necessary and Sufficient Adjustment Criteria for Identifying do- do-probability
Following Pearl [34] and Perković [37], we recall the key definitions and the associated soundness and completeness theorem.
Definition A.4 (Adjustment set [34, 37, Def. 54J). Let X. Y, and Z be pairwise disjoint node sets in causal DAG G. Then Z is an adjustment set relative to (X, Y) in G if. for any density I consistent with g
fly xx),
if 0
f(y | do(x))
V
J(y x, [(z) dz, otherwise.
Definition A.5 (Forbidden set [37]). Let and Y be disjoint node sets in a causal DAG 9 = (V,E). E).
The forbidden set relative (X, Y) is defined
Forb(X, Y. = W' E :.:W" W' De(W. f)r some W (X,YY)))
That is, Forb(X, X, Y, contains all descendants of nodes that lie on proper causal paths from X Y. Variables in this must not be adjusted for, since conditioning on them may introduce post-treatment bias.
Definition A.6 (Adjustment criterion [37, Def. 55]). Let X, Y, and Z be pairwise disjoint node sets in DAG G. Let Forb(X, Y, denote the set of all descendants in G of any W & X that lies on proper causal path from to Y. Then Z satisfies the adjustment criterion relative to (X, Y) & the following two conditions hold:
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(A1) Forbidden set condition: Z n FXrb Y, =
(A2) Blocking condition: all proper non-causal paths from X Y in G are blocked by Z.
These two conditions together ensure that adjusting for Z yields an unbiased estimate of the causal effect of X on Y.
Theorem A.7 (Soundness and completeness of the adjustment criterion [37, Thm. 56]). Let X, Y, and Z be pairwise disjoint node sets in a causal DAG g = (V,E). Then satisfies the adjustment criterion (Definition A.6) if and only if Z is adjustment (Definition A.4).
These definitions and Theorem A.7 jointly provide a complete graphical characterization for when a post-intervention quantity such f'y do(x)) can be expressed through integration over observed densities. That is, the adjustment criterion is both necessary and sufficient for identifying a single interventional distribution in a causal DAG.
A.1.2 CDE Adjustment Criteria
In the context of the controlled direct effect, this criterion (Def. A.6) remains sufficient for identifica- tion. Specifically, by letting X in Definition A.4 denote the union of the treatment and mediator sets, X (A, M), and by choosing Z that satisfies the conditions in Definition A.6, we can, according to Theorem A.7, identify the cost-intervention distribution
f'y do(a, m)) = f f(y a, m, f(z) dz,
where the integration is performed over the adjustment variables z. Hence, each compo- nent [ (a, m)] f f(y | do(a, m)) dy is identifiable, and consequently, the difference ([ m) (a" ( m)] that defines the CDE is also identifiable.
Nevertheless, it is important to note that the CDE involves the difference between two interventional expectations, E (a, m)] - E[Y(a", m)]. Because of this differencing structure, the adjustment criterion serves a sufficient but necessary condition: in certain causal graphs, neither f(y do(a, m)) nor fly do(a", m)) is individually identifiable, yet their difference- the CDE-can still be identified because -dentffiiiie components cancel out when taking the contrast. We will illustrate such cases in Figures A. and A.2.
In parallel, VanderWeele [58] proposed another set of sufficient conditions specifically tailored to the identification of the CDE. Unlike the adjustment criterion, which applies to arbitrary causal effects defined by single do-operator, these conditions explicitly target the treatment- meee--mmrdiooouuttteeeee structure and are formulated in terms of conditional independence among potential outcomes:
(B1) The potential outcomes (a, m) are conditionally independent of A given Z:
Y(a, m) 11 A Z, for all a, m;
(B2) The potential outcomes Y(a, m) are conditionally independent of M given (A,2). m). IL M A,Z for all m.
These conditions ensure that the CDE, defined as E[Y m)] E[((",,, (a", mm], can be consistently estimated from observed data, Nevertheless, similar to the adjustment criterion, they only guarantee sufficiency rather than necessity.
Counterexamples to Necessity. Next, we provide counterexamples to illustrate that Condi- tions (AI), (A2), and (B2) are all not necessary for CDE's identification. Consider the DAGs in Figure A. and Figure A.2. In the first DAG, the empty set @ identifies CDE(m) even though does not contain X,, which lies backdoor path between M and Y, thus violating Conditions (A2) and (B2). Since there is no direct arrow from to Y, the controlled direct effect is zero, i.e.,
CDE(m) = 0. When the adjustment set is empty, Equation A.1 reduces to
E(Y A= m) E(Y A= =a, m) E(Y m) E(Y m) 0,
24

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which exactly matches the true value of CDE(m). Hence, the empty set is sufficient to identify CDE(m) in this graph.
In the second DAG, the set (X2) identifies CDE(m) even though X2 E Forb(A, Y,9), violating Condition (A1). Since there is again direct arrow from A to Y, the controlled direct effect equals
zero, i.c., CDE(m)= = O. When adjusting for X2, Equation A.1 becomes
B(Y A = a, m. X2) E(Y A = a", X2) E(Y m. X2) E(Y m, X2) = 0,
which coincides with the true value of CDE(m). Therefore, the set {X2} is also sufficient to identify CDE(m) in this graph.
Figure A.1: Example illustrating Condition (A2) and (B2) is not necessary for CDE identification.
Figure A.2: Example illustrating Condition (A1) is not necessary for CDE identification.
A
M
A
M
Y
X1
X2
These examples demonstrate that both frameworks- the adjustment criterion for the do-operator and VanderWeele's s conditions for the CDE-are sufficient but not necessary. A complete graphical characterization that is both necessary and sufficient for CDE identification has yet to be established. Given that our primary focus is on the WCDE, we leave this theoretical development for future work.
A.2 Definitions Causal Partition Introduced by Maasch et al. [32]
A complete description of the eight causal partitions is included in Table A.
EXHAUSTIVE AND MUTUALLY EXCLUSIVE PARTITIONS
X1 Confounders and their proxies: Non-descendants of that lie on an active backdoor path between A and Y (Definition A.2), and their proxies (Definition A.8).
X2 Colliders and their proxies: Non -on-ancestors of {.,Y Y) with at least one active path to X not mediated by Y and least one active path to Y not mediated by A.
X3 Mediators and their proxies: Descendants of A that are ances- tors of Y. and their proxies (Definition A.8).
X1 Non-descendants of Y that are marginally dependent on but marginally independent of A (Definition A.9).
X5 Instruments and their proxies: Non-descendants of whose causal effect Y is fully mediated by A, and that share no confounders with Y (Definition A.10).
X Descendants of Y where all active paths shared with A are mediated by Y.
X, Descendants of A where all active paths shared with y are mediated by A.
Xs All nodes that share no active paths with A nor Y.
Table A.I: Partitions are formally defined by the path combinations enumerated in Table of Maasch et al. [32].
Definition A.8 (Proxy variables X1. X2, and X3, Maasch al. 32). proxy variable for X1, X2. or X3 is member of these partitions that is an ancestor or descendant of another member of its respective partition, such that the proxy not stricily a confounder, mediator, or collider, but still satisfies the allowable puth types for its respective partition (as defined in Muasch el al. [32, Table 31). This includes Xa that are descended from X3 that lie on mediator chains, X, that are ancestral X₁ backdoor paths, etc. ([32, Figure B.3]).
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Definition A.9 (Partition X4, Maasch et al. 32). Partition X4 encompasses all non non-descendants of Y that are marginally dependent on Y but marginally independent of (Table A.1). Given this definition, we observe that any X participates in u-structure A... Y + X4. This implies the following:
1. A cannot share active paths with any X4. Thus, A can share no common causes with any X4-
2. X4 is conditionally dependent on A given Y. This implicitly requires that A and Y are marginally dependent, though they may not be directly adjacent in
Definition A.10 (Instrumental variable, Lousdal [30]). Any instrument X5 meets the following criteria:
1. Relevance assumption: Xs is causal for exposure A.
2. Exclusion restriction: The effect of instrument Xs outcome Y is fully mediated by A.
3. Exchangeability assumption: Xs and Y do not share a common cause.
B Proofs and Extended Discussions for Sections and
In this section, we provide the technical details and proofs omitted from Sections 2 and 3 of the main text. We begin by establishing the sufficiency and necessity of the graphical adjustment criteria for identifying the WCDE, as formalized in Lemma 2.6. We then proceed to elaborate on semiparametric foundations, including regular submodels, score functions, tangent spaces, and pathwise differentiability. which lay the groundwork for deriving the influence function of WCDE. The section concludes with a detailed proof of Theorem 3.4, which characterizes the influence function of WCDE in terms of its identifying functional.
B.1 Proof of Lemma 2.6
Proof of Lemma 2.6. Sufficiency of the Graphical Criteria. We demonstrate that any valid ad- justment set Z satisfying Condition 2.5 identifies the same WCDE (Equation (2.1)) through two expectation-prest preserring transformations. The key insight is that the conditional independence in Criteria 2.5 allows us to safely incorporate the target mediators M' into the adjustment set without changing the identified effect. Let Z₁ = n M, Z2 Z M, C Pa(Y)) \ (M' U (A)).
Step 1: Replacing Z₂ with C Including Non-Mediating Parents of Since C satisfies the graphical conditions for identifying the CDE with mediator Z, (Definition A.3), yielding:
Ez, EE Ey (Y A a. Z,, Z } Ez, (Ez, Ey (Y A= a==";"
a", Zi, Z2)l
Ezz Y do(A a, ) B do a", Z1)l
Ez, [Ey[EBy (Y| A a, Z, C) Ez, [Ec|Ey (Y A= = a", Z1, C)]
Step 2: Incorporating Mediators in M' Define Z = Z, M We now show that augmenting the conditioning set with M' Zi does not change the outer expectation over Zi. Let W M' Zi. so we have:
Ez, (Ec Ey (Y) A = a. Zi C]] - (Y A a' a' C)}}
Bc IZ Ey ( A == 0, Z, Fc 4 4= a Z, C
Eo
Ec E Z, (By (Y A a. (.))
(EWI,,,,
A a", ", W. ]))))}}
Ec Ew (E a [
(Er A A a", Zi W.
Ee. (TZz w [ |A= W EC Ey (y a ,Z, W, C)]
Ez. ..z.w(B (Ec [E |A= 2, W, [00]]-E,,v0 (Ec (Y A= -====--- a W, C]]
= -Ez; (Ec [Ey (Y AA a. Z, C)] Ez, [Ec[Ey (Y A a² Z,, C)I}
The third equality follows from W 1 (C, A) Z in Criterion C3 of Condition 2.5, which implies p(w Z, C, A) p(w Zi); the fourth uses the tower property, and the last merges Z₁ and W
into 2 = Z U..
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Step 3: Simplification to WCDE
Then we remove elements in Z/ that are not in Pa(Y) and show
that it equal Equation (2.1).
Ez; {Eo [By (Y A a, Z C}]} Bz {Ec [Y (Y A - a" Zi, C}]}
Ec (Ez, UM Ey (Y A = a, Z, U C}]} Ec (EZ By A= Z1 UM, C)I}
EE (BZIUMi [Y A = a. M', C)]) Ee (EEZUUM [Br(Y AA=a', 1'1,',M',CN) C)]
EE (EM [y (Y AA =a.=;.; a. C}]} - -E (EM (Y AA=' dda'
EM' {Ec [E (Y AA =1==aa, M. C}]} EM' [Ey (Y) AA=',,
EM EMM(EEY [E|Y| do(A = a, aa, E do(A a", ,²))
Σ (Y do (a, m E Y do (a", m p')))((m
m'EM' WCDE
The first equality expands the conditioning sets to include M'; the second removes non-parent variables in Z without changing the expectation (all parents of Y are already conditioned on); the third marginalizes out Z1: the fourth swaps the order of integration over C and M' ; and the fifth uses the CDE identification with mediator M': since satisfies the graphical conditions for identifying the CDE with mediator M'. This yields the WCDE in Equation (2.1).
Necessity of the Graphical Criteria We now show that the adjustment criteria are not only sufficient but also necessary. By definition, The necessity of Condition C1 follows directly from the definition of the CDE, which requires blocking all mediator paths between A and Y. By definition, Condition C2 is necessary (we discuss specific conditions that could lead to necessary and sufficient VAS for CDE in Section I. We therefore focus on demonstrating that the third adjustment criterion is also necessary for given set Z to qualify as a valid adjustment for the WCDE.
Assume we have an adjustment set Z that satisfies the first three criteria, but violates the third criterion with respect to (A, Y) in graph G. Define the following subsets:
Z = Z n M,
Z = Z M. W M' Zi. C= \ (M' U (A)).
Note that Pa(Y) \ (A} is a valid adjustment set for identifying WCDE. We compare WCDEPa(Y)A with WCDEz and consider their difference:
WCDE WCDEz = (Ta(Pa(Y) \ A) - Tz)Z)) (Ta-(Pa(Y)) A) Ta- FZ)(
We first analyze the term (Pa()) (4)) - TZ(Z):
(a((((V) \ A) - (Z)
EM' (EMMA))(MM (Ey (Y A - M,M', \ (A)))
Ez, (LEZ, (E A = Zı, [2)}}
EM' (BPACY)((MPA)) (By A a, MM'M \ (M' U A))))
Ez, (EAS(Y))(MUA))
[V |A A Z, Pa(Y) \ (M' U 4))}
EM'UZ, (BPACY)AMMAA))
[Ey(Y |A= a;= M' U Z1, Pa(Y) \ (M'UA))) 4)))
Ez, (EPACY) (Ey (Y A = a, 2, Pa(Y) \ (M' A))))
The second equality holds because both Pa(Y)) \ (M' U (A}) and Ze satisfy the graphical conditions for identifying the CDE with mediator M'. The third equality follows from the fact that all parents of Y are already included in the conditioning set; thus, adding elements from Z, does not affect the conditional expectation of Y.
27

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To simplify further, we translate these expectations into integrals:
EMUZ (BPACTA(MMAA)) (Y A a, M' U Zi, Pa(Y) \ (M' U A))))}
Ez, (APACAA(MAAA)) (Y A a. 21, Pa (M' U A))}}
-1h ( uP (. p(c)dc) p(w zz)p(zz))dwe
(LC J upLy A=a,c.z)dd p(c)de) p(zz)dzı
( p (foulA-a....wiof 21)dw) du) p(c)de) p(zi)de
21
uPLUl A= p(c)dc) p(zz)dzz
( plylA=a.....,))bbVV =1) ppw A = a, c, 2, dy) p(c)de p(zz ddz
V
( ( z1)- p(w A =0=22cc))pp(eeeee)) de dz daidy
The first equality expands the nested expectations using the law of total expectation, with integration over p(w |z (((1)) and p(c). The second applies Fubini's theorem to switch the order of integration. The third rewrites the inner integral to highlight the difference. Similarly,
Ta- (Pa(Y) \ A) Ta- (Z)
w,z)lllw 21) p(w AA= = a", 21, 22, - =su 1/// (Jc (f (sowl= p a", a" c,
c)pp)))((()))) de dz dy
Subtracting the two:
WCDE WCDEz (T.PP=(TT(PA(Y))
\ \A - Tx()) -
\ A) - -Ta- (Z))
{
Z1 ( p(y A = c, w. z - -p(w A = a, 21, c)lppccppppppp/// )aw de dz dy
- -f
(J p(y A- a' c, w. a
w. 21) - p(w A = a: c))c)pccpp(2zzdd de dz dy
We can view J p(y a, w, 21) operators that map probability distributions over (C, W, Z1) onto probability distributions over Y. Arrange pvy A = w, z1) and p(y A = a", c, 21) so that this mapping is one-to-one, Then, ensure equivalence between the induced distributions, we would require
dw de dz, and f p(y A= a" , c. 21) dw do dedz as two
p(w 21 21)(())((2) = p(w
a, 21, c)p(c)p((z) p(w A= a", 21. 21. c)p(c)p((1)
to hold for all models consistently with G. However, this equality generally fails because W Kig (A, C) Zi when the third adjustment criterion is violated. Consequently, the integrand does not vanish in general, and the resulting functional WCDEz deviates from the true WCDE. We therefore conclude that the third adjustment criterion is necessary for given Z to qualify a valid adjustment set.
B.2 Illustration of Criterion C3
Example B.1 (Illustration of Criterion C3). In Figure B.3, M = [G1, B1), P G2). and M' = )G One can verify that the sets {G,, G2) and (G, B,, Gz) both satisfy Condition 2.5, and are therefore valid adjustment sets.
In contrast, the set Z (Bi, G2) satisfies Criteria CI-C2, but violates the Criterion C3. this case, Z = (Bi}, Z2 {G2), M' Z {G1), and Pa(Y) \ M' (A,G2). In Figure B.3, we
28

---

G2
A
B1
G1
Y
Figure B.3: A DAG illustrating Condition 2.5.
observe that when conditioned G1, Bi dependent on G2 due to the direct edge between them; similarly for A. Thus G₁ Mgg. G2, A B1, violating Criterion C3. One could further verify that WCDE (B,.G)) * WCDE (GDD as defined by Equation (2.2). To see this, we compare the difference between first term in WCDE Gz) and that in WCDE(G,,GG3)
T.(BBI G23) G2))
- IJ Y
If ply A a, 91, bI, 92) 91,b1,
(p)g
p(91 by b1)) dy p(gz) dga p(b1) dbj-
Because G1 G G2,Al B1, there exists a data generating process under which p(gn (b,, a) + pign (b)) the integrand does necessarily vanish (detailed construction in proof of Lemma 2.6), and the resulting functional based on {B1, G2} may differ from the true WCDE as defined. A similar argument can be made on the difference of the second term To- (1B,, -2a) (1G1, Therefore, the WCDE cannot be identified using Z {Bi, G2}.
B.3 Discussions on Influence Functions
For more detailed review of influence functions and their role in causal inference, refer the reader to Hines et al. [17], Kennedy [26], and Tsiatis [54]. Here, we focus on definitions relevant to derivations in Section 3.
To formally introduce nonparametric estimation, we begin by considering a general statistical model M, which is a collection of candidate distributions defined measurable space F2. Each element P E . admits Radon-No -Nikodym derivative p dP/ dv with respect to dominating o-finite measure satisfying 2 v-a.e. and Jq p 1. With slight abuse of notation, we use P and p interchangeably to refer to the distribution and density.
To prepare for the proof of Theorem 3.4, we begin by recalling several foundational concepts from semiparametric theory, including regular parametric submodels, score functions, tangent spaces, and pathwise differentiabillity These concepts provide a principled framework for defining and deriving influence functions. To rigorously understand and formalize the sensitivity discussed in Remark 3.3, we also introduce the definition of influence functions via local distributional perturbations. This alternative characterization is particularly useful for interpreting the robustness and local sensitivity of statistical functionals to contamination at individual points.
Tangent Space, Pathwise Differentiability, and Influence Function.
We now introduce key concepts from the local geometry the model M that play a central role in semiparametric estimation theory. We proceed step-by-step. starting with the notion of regular parametric submodels, which formalize how nonparametric distribution can be locally perturbed in smooth manner.
Definition B.2 (Regular One-dimensional Submodel, Bickel et al. 5). Let M be a nonparametric model. A regular one-dimensional submodel (Pr)ve(-s,s) - passing through P at t. 0 is a smooth parametric path such that Po = and the corresponding density Pt with respect to a dominating measure is differentiable in f at = 0, with the derivative satisfying
log (O)
E L2(P).
1=
The derivative of the log-density along submodel is referred to as the score function and represents the direction of infinitesimal perturbation in the model:
29

---

Definition B.3 (Score Function, van der Vaart 56). For a regular one-dimensional submodel {P:) with density Pt. the score function at P defined as
S(O)
log log Pr (O)
t=
which describes the direction of local perturbation of P along the submodel.
A commonly used example of regular submodel is constructed by directly specifying score function S and perturbing the baseline density as follows:
pc(o) (1 +11+t5000 tS(o))p(o),
where the score function satisfies f ((o) p(o) du(o)
and ( du(o) < oc. One can
verify that this path satisfies Po P and that its score function is
0
log (o
log + +S())) = S(o).
Øt
1=0
(=0
Such submodels particularly useful for constructing influence functions and deriving efficiency bounds in semiparametrie models.
The set of all such score functions defines the tangent space, which describes the directions in which the model M can be perturbed:
Definition B.4 (Tangent Space, van der Vaart 56). The tangent space at P is the closed linear subspace of (P) spanned by all score functions S(O) generated by regular one-dimensional submodels through P. It characterizes the set of directions in which the distribution P can be locally perturbed.
We now turn to a central concept semiparametric theory: pathwise differentiability. This notion formalizes when parameter of interest can be linearly approximated along such perturbations, and leads directly to the definition of the influence function:
Definition B.5 (Pathwise Differentiability and Influence Function, Tsiatis 54), A parameter T is said be pathwise differentiable P if there exists function tp E (2(P) with (p[}r((O)] = 0 such that, for every regular submodel {P.} with score function S(O).
T(P:)
= Erlwr(()) S(O)]
(B.1)
dt 1=0
Any such function up is called an influence function (IF) of T P.
Equation B.1 provides a constructive method for deriving influence functions, which we elaborate upon in the next subsection.
In addition to submodel-basee definitions, influence functions can be equivalently understood via local distributional perturbations. This perspective, which originates in robust statistics, is often more intuitive:
Definition B.6 (Influence Function via Local Distributional Perturbations, Hampel 14, Huber and Ronchetti 21). Let d, denote the Dirac measure at observation and consider a perturbed distribu- tion
P (1 e) E5. Then the influence function of statistical functional T at P evaluated at is defined d T. T(P:)
de
==
This definition captures the first-order sensitivity of T contamination at point o, and naturally connects to the notions robustness and sensitivity analysis. It corroborates our observation in Remark 3.3 that WCDFz is sensitive perturbations in variables within Z, implying that the choice of Z directly impacts the sensitivity the functional.
fundamental result in semiparametrie theory links these definitions to statistical estimation. Specif- ically, if T an asymptotically linear estimator of T(P) at with influence function vhr, then T is regular at P in model M if and only if T(P) is pathwise differentiable and vpp is the influence function of T. See Theorem 2.2 Newey [33]. This explains why the influence function defined in Definition 3.2 coincides with the influence function of the target parameter T(P).
30

---

B.4 Proof of Theorem 3.4
Lemma B.7. [29] Let (01.02.... 02....,0O) be a random vector with joint density po(o) under smooth parametric submodel (Pt). Assume the density factorizes as
d
pr(o) - II po,,(0; Ūi-1),
2=1
where 0:-1 (01,....00 01-1), and po,, 0:-1) = po, 0K-1) at O. Then, the pathwise derivative of each conditional density satisfies:
it po, 06-1) = po, (b-1)) Q
i
logpo,, Ōk-1) 1=
1=
Define the conditional score function So, (ōx) as the derivative of the log-density:
i
(o, = dt
log po,, 0.-1)
1=0
Then:
i Ht po o-1)
So, (o, (or Ō;-1).
t=0
Moreover, under the chain rule for scores, this can be written in terms of the full-data score function S(O) as:
お. Po (0; 0i-1) t=
= (E[S(O) O, o.] E(S(O) 0 = =ō.-1)) po, (ox Ōz-1).
This lemma allows for expressing the derivative of marginal or conditional density in terms of conditional expectations the full-data score function and is essential for decomposing pathwise derivatives in IF calculations.
Proof. (Proof of Theorem 3.4)
For the WCDE functional:
(Z) Fzz ZE
E|Y
A a, Z,, Zz])
Recall definition of the pathwise derivative:
8 T(())
(o; P))(o)p(o)do.
0l
where the score function S0) is defined as
i
S(o)
log Pr (o)
Øt
1=0
for any smooth parametric submodel {Pt)-
We start by expressing Ta (P.) as:
/ upi(y a. a. Z2) (z)ddzzzzd
T. (Pt)
/ pilys 21, 22, a) (Z)) dydzzdzz
Pe(21, 22, a)
Taking the derivative w.r.t. at t = 0, and applying the product and chain rules, we get: 0T, (Pt)
t=0
1
p (z)) 1p)p p (zz)
0
(.. a, 21, 32 (zz) a + p (a, 21, 22)
Pt (y. 21,
Ot Pt (zz)
p (a, Zi, m2) 01
1
first
term
p (y, a, Z1, z2) (22)
( a, Z1 2z p (111 (zz)
+
0r Pt (zz)
Pt (a, 21, 21,22)
de(v) du(zz) du(zi)
p (a, 21, 22)
p (a, 22, 21 )²
t=0
(ii) third term
(iv) fourth serm
31

---

Each term is handled individually. For the first term:
11 (21) p (22) a P (1, 21, (2)
Ol Pt (y. a, 22)
jdv(())dr(())ddutz))
t=
/ vl p (Z2) (z1) Pt (3, Ze
0
logp: (y, a, 21, 22)
paa 21,
d1.
===0
/ vl p (21) (22) p(o) ((o))du(())d((22ddviaa
(a, Z1, 22)
/ p ((1 p (22)
La (i
(a, 21, 22)
/ La(a)) (zi) p (22) ((o)p(0))db(())
p (a, Z1 b2)
In the third equality, we extend the integration from the conditional domain over (y. Z1, Z2) to the full domain (y. a', 21, z2). To preserve the original conditioning on A a, we introduce the indicator function La (a') == [a==a), where denotes the fixed treatment level of interest, and a' serves as the input variable to the function, which may vary throughout the integration.
For the second term, we apply Lemma B.7. This result allows us to express the influence of the perturbation in p(zz) on the target parameter through a conditional expectation of the score function.
f p (v, a, 21, (21)
a Ot Pt (z2) t=
(v(y)dv(z))dd(e)
p p(a, Z1,
22)
(B[S(O)|z)] E[((()] p(zz) du (z())ddo(z)dd((z)
p (1, 22
We now expand the conditional expectations explicitly. The first term inside the integrand is rewritten using the definition of conditional expectation and the joint density, while the second term represents the overall mean of the score function. By computing the expectation of Y given (A a, Z₁ Z2) and rearranging the integration order, we ultimately express the entire term as an inner product of the score function S(o) with centered function depending Z₁ and Z2, which contributes the influence function.
p (y. a, Z1, 22) (21)
(E[S(0))Z2) - E[S(O))) dv (d)y)dd(())dd((z)
p(a,z1 Z1, Z2)
p (B, a, 21, 22 (Z1) p(zz) (ʃ
p ( (2,222,21
f ((o)p(o)du(o) du(y)du(za)dv(z)))
/ Ex Ey 22)1 (f
Slo)p(u... bidd
/ ((o)(()de(()) du(z))de(z1)
/ Ey (Y (f I=(P)5(())po)dr())
22)d)/u..zzz2)))dd)vvvva) dz(z))du(Zz)
/ (Ez, [y Zi Zoll ( (i du(zz)
If [I((P)S(o)p()drr(())
(Ez, [Y A a, Z,, [o]l] (8(0)((o)ddu() -0000)/000000/()SS000000000-- T.(P)S(o))(()duo
32

---

For the third temn, we apply Lemma B.7. This result allows us to express the influence of the perturbation in p(zi) on the target parameter through conditional expectation of the score function.
f P (y, (2) a
} du duly)de(a))dd(zz) (du(2)))
0t p (21)
p(a, 21, 22)
= f p (V: 21, p (Z2)
(B[S(O)|z]] - p(z1) (z())d((zz))dr(z)
p 21, 22)
The rest of the computation follows by evaluating the conditional expectation of Y given A a, 2,, Z2 and expressing the result in terms of an inner product between the score function and an identifying function.
(1) a, 21, 22) (22)
E[s(O)) (otl (z())dd(()))dr(z))
(aa 21, (2)
(y, a 21 22 (Z1) p(zz)
(ʃ
S(o)p(.. a, (
(1,21,222 22)
-f ((o)p(o)dr(o)) du dr(w)dd((s)dd(zz))
- (x(
(J S(o)p(y, (o
(Y 21, 21, 22)
/ ((o)p(o)dr(o) du(zz)du(zz)
/ (y (ʃ s(o)plu, a, (s)dr(())dd(((drr(z)) du(z))du(zz)
/ Ia(()S(o)pp()drro))
/ (Ez, [Br[Y Z1, Zelll (j
du(zi)
(T(PP)S(o)(()dd(o)))
(Ezz Ey Y a, Zi, 22]]] ((0))(())du(()
For the fourth term, we begin by applying Lemma B.7 to express the derivative of the joint probability Pe(a, 21, 22) in terms of the score function S(O).
f
(3, a, Z1, 22)
θ de Pt (a, Z1. 22)
dv du())du(2)))dr(zz)
p 22,
ʃ y ( p1)))))) (E|S(O))a, 21, zz - E(s 21, 22) ) (22))))(2)))d((2)
p p(a, 22)
Next, we expand the conditional expectations as integrals over the relevant conditional distributions. The term [O)(a) Zı, 22] becomes integral over p(yla, Z1, z2), while E[S(()) integrates over
33

---

the full joint distribution p(o). We then factor out the conditional expectation of Y given (a, Z1. 22).
= = U. a, 21. 22) ( (Z2)
(E]S(O))a,z) 21, 22] EIs.
du (e(s)du(zs)du(a))
p p(a, 21 22)
f p(y, a, 21, 22) (z)) (z2)
( 210, - S(olplondu(o)) ((())dd(s))d((
(1,21,22)
/ (p(z))((z))BB By (Y 21, x2))
(f 20,0pl
((o)p(o)du(o) du(z))du(zz)
= (1
(f
S()) ply, plu,a,21,22). du(v) dv(z) )dv(2)) pia, 21, k2)
[a(P)S(o)p(o)droo))
Finally, we introduce the indicator function la'a') and combine all integrals into a single expectation with respect to p(o).
/ (p BY 21, 22)3 (f plu Pl) du(u)) du(z))du(22) - / Ta(P)Sto)p(ojarl)))
p(a, 22)
/
plea)p(ze) Ey (Y 1a, 21, ml) (1
S(o)o(o)de(s) du(zi)du(zs)
Ta Ta(P))(o)p(o)dolo)
p(a, 21. 222
/ La )p
By (Y a. 21)
p(a, 21,
(a(a')((z1))) pp p(a, 21, 22)
-1ooo/)doooo - 7TTISS0OOO0)) 3
Ey Ey 1a, 21, 22)
Summing all terms together, we obtain:
aTa (P.)
Ot
=/ [1 la 1=
[ Ie(a') (21) (22) (y (
By (1
+ZZ [Ev (Y|A a, 22
p Pa,,z,, 22, 21)
Rzz Br (Y|A 21,
= / Valo: (o; P)S(0)((o)dd(()
with
Ta [a(A)p(Z,)) (pZZ))
(Y (y-Ey Ey Y A a, Zz, Zzl
via (Y, A, Z1 Zz;
p(Z, Zzz a)
Ez, EY A a, Z₁ Zz]
+ EZ, Brly AA a, Zr. Zz]
2P)
required.
C Proofs of Results in Section 4
Theorem 4.3 follows from the four lemmas presented in Section 4. Accordingly, first provide the proofs of Lemmas 4.4, 4.6, 4.5, and 4.7 individually, and then proceed to prove the theorem.
C.1 Proof of Lemma 4.4
Proof. (Proof of Lemma 4.4: Supplement of backdoor variables)
We begin by showing that the (G1, G2, B2) remains a VAS. We can verify this by comparing the identified WCDE functionals, Recall that for any adjustment set Z (Z), Z2), the WCDE is identified as
WCDEz = Ezz E2o
[E2 [Y A= a, 22) Ez [ A a", Z, k
34

---

B₂
G2
G2
A
B
G₁
A
Gi
Figure C.4: A DAG illustrating Lemmas 4.4, 4.6.
Figure C.5: DAG illustrating Lemmas 4.5, 4.7.
For (G1, G2, B2). consider the term corresponding to
(G 22 B2) = EG BG B(Y A= G1, G2, B2)
Under the conditional independence
G2 و A, G. B2
(C.1)
have p(G,, B2) iterated expectations,
p(G2 B2)p pGG A a, G1 (2)) and by the law of
EG, B2 [E A a, G1, G2, B2) E(Y A = G1, Bz).
Substituting this gives
(G G2, B2) EG, EB: Ec, n. E[Y A G1. G2. Ba)l]]
Ec, En, [[(Y A a, G1, B2)] TT (G1, B2),
and the same equality holds for a", Hence,
WCDE
By the uniqueness of the identified WCDE functional, any adjustment set that yields the same functional VAS must itself be VAS. Therefore, (G1, G2, B2) is also VAS.
now compare the asymptotic variances of two VASs, (G), and (G1, G2, B2). For clarity, we first focus on the first component of the WCDE in Equation (2.2), and let o (P) denote its asymptotic variance. The extension to the full WCDE is straightforward. We begin by decomposing the variance:
2 (,,(C.,B)(P)) Varp (Y, A, G1, B2; P)] Varp (V, AY G1, G2, B2; P)]
+ Varp [tba (Y, A, G1, B2; P) (Y,Y,, A, G1 G2, B2; P)]
2 Covp (V, A, G1, G2, By; P), (Y, A, G1, By; P) (Y, A, G1, G2, By; P)). To analyze the covariance term, consider the difference in IFs evaluated (Y, A, G2, G2, B2): YY, G1, B2; P) Va, G1, G2, B2;
(a(A)p(G G. )P(B2)
Y - Ey [Y -y
A= a, G1, B2] + +EG.
GG,,, B2,a) a)
+ EB32
Ey Y A = G1, By]
LY A a, G. By
Ia(A)p(
G1 )pppppppppp G2, B²)
Y (r-Ey Y
A= = G1, G2. B₂]
p(G,, G2, B2,a)
E...B. [Y A= G1, Gz, B₂]
- EE By Y A a, G. Gz, Bz]
Ia(A) (Gn): (((2)
Ey Y A= G1, G Byj Ey Y A= a, G B2)
p(A= a, B2)
EB. Ey Y A G B₂]
+ EEG.
Ey A = G. B,
E...B,
Ey Y A= a, G1, Gz, B²]
EG, Ey Y A a, Gr, Gz, Bz]
35

---

The second equality above holds under the conditional independence assumption C.I, from which it follows that
p(G² B₂) = p[ G1, B2, A = a))
which implies:
p( G2, (1)
PG2 B2) GG B2, A= p(G B2, G1, A=
(B2)pp(Gi) (G,)
p(2)) p(A= a, G₁, B2)
pl G1, G2 a=
Therefore, the first terms in the two influence functions cancel out. To simplify the final term, note that:
BGg.B2
Ey A a, G1, G2, B2]
EB2
EGA/B=
By Y A=a, G1, G2 B₂]
Ey Y A a, G1, B2] this equality follows from the conditional independence assumption C.1 and the law of iterated expectations. Substituting back, the difference simplifies to:
-EB, BG Ey A a, G1, G2, B2]
EB,
(A) p(G.) p(22) p(A a, B2)
(Ey EB, Ey Y a, G1, By]
A a, G1. G2. B₂] Eyly A G1, By]
+ +EG
Ey Y A a, G1, By]
EG..B, B. |Y A= a. G2 B₂]
EEG Ey A a, G1, G2. By]
Ia(A) (G1) (((2)
Y == a, G1, G2, B₂] R a, G1, B2]
p(A=a, a, G1, B₂)
+ EG, Ey Y A a, G1, B2] EG Ey Y A= a, G1, 22, Bg]
Therefore, we arrive at the simplified expression for their difference.
We then construct parametric submodel that perturbs only the conditional law P(A, Gi G2, B2). while keeping all other components of the distribution fixed. We show that under the conditional independence assumption C.1, the difference between the two IFs is orthogonal to the influence function of the larger adjustment set:
Ep [Y, A, G1. G2, BN: P)- (Y, - &. (Y, G1, G2, B2; P))] = 0.
Hence, the covariance term vanishes. Substituting this into the variance decomposition yields:
Varrlva (Y, A, G,, P2;
Varplya (Y, G1, G2, B2; B)] (arp/
G1, G2, B2; P)].
where the second term is nonnegative. To formalize this orthogonality, we appeal to two key results from semiparametric theory [54]:
1. If P, is smooth parametric submodel with Po P and score function g. then the influence function th satisfies:
d (P.).
= =E g)-
dt
2. In a nonparametric model, any function such that Eplg(A, G1, G2, B2) G2, Bg] = is valid score function for the conditional law P(A, G Gg, Bz).
36

---

Let us now consider any distribution P and define the function
g(A, G1, G2, B2) (Y, A, G1, B2; (Y, G1, G2, B2;
which satisfies the second condition above. Using this function, we define the following submodel for sufficiently small 2 0:
PY A, G1, G2. B2) P(Y A, Gr, G2) P,(A, G1 G2,Bz) P(G2, B2),
where
P. Gi B2) P(A, G2, B.) ((A, G1, G2, B2)).
This submodel perturbs only the conditional law of (A, G1) given (G2, B2), leaving the remaining components of the joint distribution fixed. The score function of this model at t is
log pr G1, B2)
log(1 +t g(A, G1, G2, (2)3
= =A(A, G1, B2).
t=
1=
Applying result (1), we obtain:
d
Ta (P:)
=Eply g].
di
t=
Now consider the functional:
Ta(P) = Σ
Σ G1 G2.B2
BY a, G1, G2, B2] P(G,, B2)p(G)).
This functional depends only on the joint distribution of (G2, B2), G1, and the conditional distribu- tion P(Y A, G1, G2, B2). If the function g satisfies
Eplg(A, G1, G2. B2) G2, B2]=0 =0 and Epl((A, G1. G2, B2) G1] = 0,
(C.2)
then perturbations in P,,Y, G1, G2, B2) do not affect those distributions. That is, the submodel is locally ancillary for the functional Ta, and therefore:
d
- To (P.)
= 0.
di
1==
Combining with the earlier result gives:
E Ep(pagl =
Then obtain the orthogonality:
Ep [vb (Y, A, G1, G2, B2; P)g(A, G1, G2, Ba)]
We now verify that this function g(A, G1, G2, B2) satisfies the two orthogonality conditions in Condition C.2. First, consider the conditional expectation given G2, Bg:
Ep(ea (Y, A, G1, B2; P) bY, A, G1. B2; P) G2. B₂]
[a(A) (B)) ((,))
Ep
[ErlY A = a, G1, G2, B₂] Er A a, G1, Ba)
p(A a, G1, B2)
EG,
(Y AA= a, G1. B2]
- EEG,
Ey Y A G1, Gz. B₂]
Gz, Bg
pG2,, B2) (())
Ep
EylY A G1, G2, Byl - Ey Y A a, G1, B2]
(G,, G2, B₂)
+ EG,
BY A = G1, G1.
- -EG, Ey A a
G2, By
G2, Bg
p(Gz, B2) p(G) G(G G2, BB)
Ep
Ey(y A= G1, G2. By] - Ey Y A= G₁, B2)
Gg, Ba
+ EEG
Ey A = aa, G1, B₂]
- -Bai Y A = G1, G2. B2
37

---

In the second equality, we integrate over A conditional on (G2, B2). Note that the last two expectation terms are fixed once B2) are given, so we can move them outside the outer expectation accordingly.
We now rewrite the first expectation integral over 11 conditional on (G2, B2):
pl 22. B2) ((GG),
[v A= a, G1, G2, B2] By A a, G1, Ga) G2, B2
Ep
p(G1, B2)
p (22 B2) p(g))
(By [Y] == a,g1 G2, B2] Ey Y = a, g1, B2)) G2, B2) dgi
p(gı. G2, B2)
= =/(Br (Br|y ( A = g1: 22, B₂] Ey Y A = a, g1 B2)) ((g) dgı
EG. Eyly A = a, G1, G2, B2]
EG, Brly A = a G1, B2]
Substituting this expression back into the original conditional expectation, we obtain:
Eply (Y, G1, B2; - G1, G2, By; P) G2, B2]
= EG,
Ey YY]
A = G1, G2, B2]
EG, Ey Y A = G1 B2]
EG
Ey Y A= = a, G. Be]
EG, Ey Y A= a, G1, G2, Bu]
0
This establishes that the function g(A, G1. G2. B2) satisfies the first orthogonality condition in Condition C.2, Ep[g((A G1, G2, B.) G2, B2]B2]=0 =0
To verify that it satisfies the second orthogonality condition, we proceed similarly by conditioning on G
Ep Er A, Gr, B2; P) (a, (Y, Gr, B2; P) G1)
= =Er [a(A) ((G ((..))p((B))
By(y A B a, G1, G2. B₂] - Ey Y 4 a, G1, B₂)
p(A = a, G.
Eg,
Ey Y A= G1 By]
EG, Y A a., Gz, By
G
(G,) PGG B2). G(G Go, B2)
Ep
EY A a, G1, G2, B₂] - Ey YY A= a, Gi. Bz]
Ec,
Er AA=a G., B₂]
- EG, Er A a, G1, G2, B2
G
38

---

In the second equality, we integrate over A conditional on G1. We now rewrite the first term as an integral over (G2, B2) with respect P(G2, B2 G1), and then simplify.
(GG) p, p B2)
Ep
[Ey|Y
A == a, G1, G2, B2] By A= a. G1, G2)
EG,
By |A G1, B₂]
- EG, By Y A a, G1. G2. Byl G1
p( G(G p(( G2. B2)
[Ey A 0, G1, b2] - l A = G1, bz)) p(g2, b2 G1) dg2 ab2
G(1,, 82; b2)
+ Ep EG: Ey [Y]
A= = a, G1, By]
EG:
Ey L A = a, G1. G2. B2]
G1
p(g.. b2) (Ey Y A = a, G1.82, - |y A = G1, bal) dg. db2
Ep EG Ey Y A a, G1, Gr By] -EG, Ey Y = a, G1, G2, B2]
G1
Next, cancel the two components of the expression by invoking the conditional independence assumption C.I.
/ p(g., b2) (BrlY A- = GI, b2] 1 Er(Y A = =a, G1, b2]) dga db2
Epp EG. Er[Y A a, C1, EG. Ey [Y A a, G1, G2, B2]
Gi
= ff (By|x =(BBY/X | AA= a, G2,g2, g2, b2] - YY A= a, G1. bol) p(ge b2, = a, G1 G:)p(b2) dg2 db2
+ Ep Bg. [Y A = a. G.. - EG.
lY A a G2, G2, B2]
Gi
Ep EG,
Eg,
Ey Ey Y a, G.. Bg] EG,
By By Y M A= a. G1, G2, Bg]
G
The second equality holds because we have the conditional independence assumption C.I, so: =a p(g2, b2) = p(ga b2) p(b2) = p(g2 b2, A= a, G,)p(b2), which justifies replacing the joint density in the integral. Furthermore, by the law of iterated expectations,
f By By G1. ba) p(g2 A- G.) dg: = E a, G b2). As result, the two integrals cancel out, yielding the final equality. To simplify this expression, compute both terms as iterated integrals.
Ep EG, EY A a, G1, B2] EG, Ey A= a, G2, G2. Bz]
G1
= -slʃ v. pll A = a.g1, p(p dgi p(b2 Gi) db2
- -sllʃ y. iply = a.gg.... bz)) dy p(g:) dg. p(ga, by G1) dg: db:
39

---

Based on Condition C.1, we can simplify the second expression through the following derivation. First, we apply Condition C.1 (p(g2 b2) = p(ge
a, G1, b2)) expand the conditioning set:
||]] y (yl a, B1, 82, ba)l dy p(g)) dgi p(g2, b2
Gi) dg2 db2
= -[]l] y ply A = a. g1 82. b2) d dgi
p(ga by)dge p(b2 G1)dba
y ply A a. g1 82 b2) dy dg. p(g2 A a. g1, by dg)2 p(b2 G1)db2
Next, we express the integrand as a joint probability distribution over y and g2:
[]ʃl y p(y.g2 82 a. a.gı, b2) dydgæ p(g:)dg: p(b2 G1]db;
Using the chain rule of probability, we factorize the joint probability:
/|ʃl p(ge = a, b2) A a, g1, b2) ((g)) p(b G1)dba
Finally, we marginalize over g2 by integrating it out (since J p(gz y. A = a, g1 by)dg: = 1), yielding the simplified expression:
fIʃ ply A a, g1, ba) d(p()))d)))
p(b,) G1)db2
This shows that the second term equals the first one. Thus, we conclude:
Gi ]
Ep Eg, Eyly A a, G1, G2. Eg, By [Y | AA a, G1, G2, G2,
So far we have shown that the function g(A, G1, G2, B2) satisfies Condition C.2, so we obtain the orthogonality:
Ep (Y, A, G1, G2, P) (vo (Y, A, G2, B2; P) (a A, G1, G2, B2)] = O.
Therefore, we conclude that the variance decomposes as:
2 o.) varr [v, A, G1, B2; P)]
= varp (Y, (Y, A, G1, G2, B2; P)] + varp (t (Y, A, - - G1. G2, B);
Now extend this the vectorized WCDE. Define: c (Ca)aEA, Q (Qu)aA where
Qa v. (Y, A, G, B2; - - (Y, A, G1, G2, B2;B1P P)
For any Z. define A, Z; P) (va (Y, A, Z; P))GEA. Then, writing EasA Co 1. A, A.P)) =
c (Y, A, Z; P) We notice that:
[Q
G2, Bg] = 0, Ep[Q G1] = 0,
Then we obtain the orthogonality:
Ep Q (Y, A, G1, G2, By; P)] =
Therefore, we conclude that the variance decomposes as:
d...B.)) P)= = varp [c (Y, G1, By; P)]
varp cc (YY, G2, By; P)] + varp [c [Q] (P))) +cT - varp(Q)c
This completes the proof.
40

---

C.2 Proof of Lemma 4.6
Proof. (Proof of Lemma 4.6: Deletion of overadjustment backdoor variables)
We begin by showing that the set (G1, G2) remains a VAS. We can verify this by comparing the identified WCDE functionals. Recall that for any adjustment set Z (Z,, Z2), the WCDE is identified as
WCDEz Ez, [ A Z1 Zz)] Ez, B(Y A a² Z1. 32
For (G1, G2, B2), consider the term corresponding to
Ta (G G2, B₂) EG
E B, A a, G1 G2, Bz)]
Under the conditional independence
Y l B2 G1, G2,
(C.3)
the inner expectation simplifies
E[Y A = a. G1, G2, B₂) = B|Y A = a, G1, G2),
Substituting this gives
Ta (a B2) = EG, EG.,B, B(Y -A= G1. Gz)]
Ba, Rc, [YY A G1. G2)] = Ta (G. G2).
and the same equality holds for a". Hence,
G., = WCDE
By the uniqueness of the identified WCDE functional, any adjustment set that yields the same functional VAS must itself be VAS. Therefore, (G1, G2) is also VAS.
now compare the asymptotic variances two VASs, (G1, G2) and (G,, G2, B2). For clarity, we first focus on the first component of the WCDE in Equation (2.2), and let o ( denote its asymptotic variance. The extension to the full WCDE straightforward.
We begin by applying the law of total variance to decompose the variance under the larger adjustment (G1, B2): Varlpa A, G1, G2, By; P)] Var P (Y, A, G,, G2, Bz; P) A,Y, |) G1, Gz]] + E Var va (Y, A, G1, G.2, B2; P) A, G1, Gz]] Next, we will show that under the conditional independence assumption C.3 the based on adjustment set (G1, G2, B2) satisfies
Ep|ya (Y, A, G.. G2, B2; P) A,Y, Gz] = (Y, A, G2, Gz P).
This equality allows us to relate the variances of the two IFs via the law of total variance. We now compute the conditional expectation of dia (Y, G1, G2, P):
Ep [Y, A, G1, G2, B2; P) A,Y, G1, Gz
((G p G2, B2) p(A a. G G2. B₂)
(A)) (Y- Ey Y A G, G2|) Ep
A= G2
( first term
+ Ep EG,,n, B2 Er 1== a, G1, G2, Ba]] G
second
+ Ep EG, Ey Y A a. G Gy, Bzl G2 -T..(P)
(ii)) term
41

---

This equality holds because, given the conditional independence relation C.3, it implies the key simplification: (y A a, G1, G2, B2) = By A a, G1, G2)
For the first conditional expectation term, we perform the detailed computation:
GG2, B2) p P(A, G1, G2, B2)
Ep
A a, a, G1, G2
((G p( G2. B2)
Ep
A A= a, Y, G1 G2
p B2 A 0, G1, G2)
((A = a, G1, G2)
((G) (G2, B2)
Ep
A= a. Y, G1, G2
p(A = Gi, G2)
(B2 A A= a, Gi, G2)
((G)).p(G2. B2)
Ep
A= a. Y, GI, G2
r(A = G1, G2)
(B2 A 1=a,60 a, G1, G2)
p(G,) (G,,,, by)
(p(b2 = a, Gi G2) db2
p(A = 0, G1, G2)
p(b2 A = a, G1, G2)
p(ba
A 0, Y. Y.G, G2)
= p(A = a, G1, G2)
p(G.) p GG2, by)
db2
p(b2 A = G₁, G2)
In the last line above, we expand the conditional expectation as an integral over B2. Now, under the conditional independence assumption C.2, it follows that
p(b2 A a, Y, G G2) - p(b2 A- a, G1, Go).
Substituting this into the integrand, the ratio simplifies to 1:
pbb A = a, Y, G1, G2)
b2)
b2) .(G,).
dbg
P(A = a, G1, G2)
p(b A = a, G1, G2)
p(Gg, p(G) db2
p(A G1. G2)
b2
1 p(A= = G G2)
P[G₂ b2) p(G) pl G2) db2
bb
p(G.) (G2)
p(b2 G2) db,
P(A G1. G2)
bb
( (G2)
p(A a, G1: G2)
Since Ey (Y
A = a, G1, G2, B₂) = (Y A= G1, G2), the second and third terms simplify
as follows:
Ep
D, A= a, G1, Go, B2] G
= =Ep EG,, Ey YY| A = a, G. Gol
G
EG.
Er A= a, G Go
Ep EG. Ey Y A= G,, G2, By] G2
=Ep EG, EE Y A = G1, G2]] G2
E., Y A = a, G. Gz]
42

---

Combining these results, we obtain:
Ep [va A, G1. G2, B2; P) A,Y, G,, Gol
Ta(A) ((.) p(G2)
Y - Byly A= a, a, G1. Gzl
p(A = 0, G1, G2)
EG. Ey Y A a, G1, Gu] d (Y, G. P)
+ EG, E A == G1. G2l
PP)
That is, the conditional expectation of the influence function under the larger adjustment set equals the influence function under the smaller set.
Finally, applying the law of total variance, we establish the variance dominance relationship:
o Go. = varp It (Y, G1. B2; P)]
= var [1 (Y, G1. G2; + Ep [varp
G1. (e, A,Y, G1, G2]
2 a (.)()
The inequality follows because the conditional variance term is non-negative.
Now for the full WCDE estimator, define: с (Co)aEA, for any Z, define (Y, A; (Y, A, Z; P)) Then, writing EGEd (Y, = c A,
note that:
Ep [c A,, G2, B2; A,Y, G,, G2] =
T (b, A, G1, G2; P)
So again by variance decomposition:
(P) G...,...,)PP)) varp [c bb(Y, A, G1, G2, B2; P)]
varp|Ep [c (b, G1, G2, B2; P) A,Y, G1, G2]] + Ep [var [c &Y, G1, G2, B2; P) A,Y, G1, G2]] varp cc (Y, A, G1, G2; P)] ce Ep [varp (b, A, G1, G2, By; P) A, G1, G2]] =., (P) Ep [varp (T, A, G1, G2, By; P) A,Y, G1 G2]]
2 oo.
This completes the proof.
C.3 Proof of Lemma 4.5
Proof. (Proof of Lemma 4.5: Supplement of mediator variables)
We begin by showing that the set (G1, B1, G2) remains a VAS. We can verify this by comparing
the identified WCDE functionals, Recall that for any adjustment Z (Z), Z2). the WCDE is identified as
WCDEz Ezz Ez, [B(Y A Z1, EZ, [B A a- Zi Z1 Zz)3
For (Gi, B1, G2), consider the a-term:
Ta (G,, Bi, G2) EG,.B, EG, B(Y A a, BI, Ga]]
EB, EG B,
Ec. E Y 4= G. Bi. G
By the conditional independence assumption:
G وg A, Go B
(C.4)
43

---

we have p G1, B1) p (1))(B:) - p(G, A = 22, Bi)p(BB... Applying the law of iterated expectations
|B,
c.
A= a, G B1, Go]] Ec. |B, A = G. B1, Gz)l
EG.
Eg, A= B(Y
A a, G1, B1, Gz)l
EG,
[E(Y
A == 0, B1, G2)]
Therefore
G., G2) = EB, EG A= a, Ga) (BI, G2),
and the same equality holds for a". Hence,
(C. = WC
By the uniqueness of the identified WCDE functional, any adjustment set that yields the same functional VAS must itself be VAS. Therefore, (G), B1, G2) is also VAS.
We now compare the asymptotic variances of two VASs, (B1, Gz) and (G,, B1, G2). For clarity, first focus on the first component of the WCDE in Equation (2.2), and let o oz (P) denote its asymptotic variance. The extension to the full WCDE is straightforward. We begin by decomposing the variance:
2 G2) (P) = Vary (Y, B1, Gy; P)] = Vary [ (Y, G1, B1, B1, P2;
+ Varp [Y, A, B1, G2; (Y, A, A, G1, BI, G2;
2 Covp (Y, G1, B,, G2; P), v. (Y, A, B1, P2; dh a. (Y, A, G₁, B1, P2;
To analyze the covariance term, consider the difference in IFs evaluated at (Y, A, G1, B1, G2):
A, G22 P) (Y A, G1, B1, G2; P) [a(Ap((B )p(22)
Y - Ey -r Y
A a, B1, G2]
p(B,, G2.a)
EG,
Ey Y A B1: G2]
+ EB
Ey Y A a, B1, G2
La(A)p(G B (1)p((G)) (G2) p B1, G2. a)
Y Yr (Y
A = G B G2]
-Ba,
Ey Y A a, G1, B1, G2]
EG y A a, G1, B G2]
Ia(A) pB)) p( p(2)
Yy a, G₁, B G2] Ey A= a, Bi, G2]
p(A a, Bi 22)
Ea,
Er A a, Bi, G2]
+ EB.
Ey [Y
A = a, B, G2
E..
By Y A a, G B1, Gzl
EGGB Bj
By Y A= a, G Bj, G2)
The second equality above holds under the conditional independence assumptionC.44 from which follows that
p(G B₁) = p(G B1, G2, A = a),
which implies: p( G.
BBB))p(C p(G G,, Bi, A= (Bi, G2, A= G Bi, G2, A= p(A= a, B, Go)
BB (p(2))
G B
Therefore, the first terms in the two influence functions cancel out. To simplify the final term, note that:
EG,,B Bi
By G1, Bi, G2] EB, EGG|B
Er A a, G.. Bi. Go
EB,
Ey A = G Bj, Gz] C2
= EB,
Ey Y = a. Bi G2
44

---

this equality follows from the conditional independence assumption C.4 and the law of iterated expectations. Substituting back, the difference simplifies to:
[A)) pBB)) p(G2). p(A a, B1, G2)
(BylY
A a, G1. B1, Gy] - By A a, B1, Gyl
BG.
By A a, B1. G2]
+ EB.
Ey A a. 1 G₂l
B..,
| A a, G1, B1, Gy]
EB..G
Ey A a, G1, B1, B, Gzl
Ia(A) (B1) p( (G2)
By A a, G1. B1. Ge| Ey A= a, Bi, Gzl
p(A = a, Bi. 22)
BG. Ey Y A a, Bi. G2] EG, Ey YY A= a, G Br Gzl
Therefore, arrive the simplified expression for their difference.
We then construct a parametric submodel that perturbs only the conditional law P(A, G2 B.). while keeping all other components of the distribution fixed. We show that under the assumption C.4, the difference between the two IFs is orthogonal to the influence function of the larger adjustment set:
Ep [ (Y, G1, BI, G2; P) (1%, A, B1, G2; P) (Y, A, G1, BI, G2; P)) 0.
Hence, the covariance term vanishes. Substituting this into the variance decomposition yields:
Varplya (Y, A, B,, G2; P)] A, G1, B1, G2; P)] Varr (Va (Y, B,, G2; P) v)((,, A, G1, B1, G2; P)),
Var Varr-(ya (M,
where the second term is nonnegative. To formalize this orthogonality, we appeal to two key results from semiparametric theory [54]:
1. If Pt is smooth parametric submodel with Po = P and score function g, then the influence function tha satisfies:
d To (P.)
= Eplva gl-
dt
1==0
2. In a nonparametric model, any function such that
Ep GT, B1, G2) G1 =0 is valid score function for the conditional law P(A, G2 Let us now consider any distribution and define the function
G1,
g(A, G B1, G2) (Y, G2; dh (Y, A, G1, B,, G2;
which satisfies the second condition above. Using this function, we define the following submodel for sufficiently small 2
P. A, G. B,, G2) P(Y A, G.G.B., Gz) (AA.G G B,) P(G,,B.),
where
P(A, G1, B,) ((A, G1, B1, Gz)). This submodel perturbs only the conditional law of (A, G2) given (G1, B1). leaving the remaining components of the joint distribution fixed. The score function of this model at t is
P/ G1, P(A,
log A, Gr, Gr, Gz)
log + ((A, Br, Gz) Ōt
=g(A, G₁, B1, G.).
t=
1=
Applying result (1), we obtain:
d
T (P.)
Er Erlaa gg).
dt
t=0
45

---

Now consider the functional:
(P) - ΣEEB
By a, GI, Bi, Gell B, G2
This functional depends only on the joint distribution of (G). Bi), G2, and the conditional distribu- tion P(Y A, G1, Bi, G2). If the function ! satisfies
Ep G1, B1, G2) G1. G1
and Eplo(A, G1, B1, G2) G2] = 0,
(C.5)
then perturbations in P:(Y, A, G1 B1, G2) do not affect those distributions. That is, the submodel locally ancillary for the functional T., and therefore:
d T... O. di
Combining with the earlier result gives:
Er(Pag]
Then obtain the orthogonality:
Er(va A, G1, BI, G2; P)a(A, G1, Bi, = =0 We now verify that this function g(A, G1, B,, G2) satisfies the orthogonality conditions in Condition C.5. First, consider the conditional expectation given G1, B1:
Ep(va (Y, A, B1, G2; P) (Y, A, G1. B1, G2; P) G1. B1]
Ia(A) (p(2)) (B.)
Ep
[Vy|V A a, G1, B1, G2) -y)-Byyy B A a, B1, G2)
p(A = a, B1, G2)
+ EG,
By | AA a. B1, Gzl
EG, Ey A a. Gr Bj, Br G2l
G1. Bj
P B()))()))) pG2))
Ep
Ey[y A a, G1, Bi, Gz] Ey Y
a, Bi. G2l
p(G,, Bi, G2)
Ec,
By Y A a, B1, Ga]
Eag Ey Y A a, BI G2
G1, B
Ep
p B1) p(2)) p(G,, B1, G2)
Eyly A a, G1, B1, G2] - Ey Y A= Bi. Gal)
Bı, G1
EG,
Y A B1, B1 G2l - EEG Ey a. G. B1, Gzl
In the second equality, we integrate over conditional on (G1, B1). Note that the last two expectation terms fixed once (G1, B1) are given, accordingly.
them outside the outer expectation
We now rewrite the first expectation as integral over G2 conditional on Bi)
p(G2) p(G,, B,, G2)
Ep
|y A a. G.. Gal - Er A B., Gal) G1, Bi
= / PG,, B1) p(g))
(Ey Y a, G₁, B g2] Ry ( a, B,, B2) p()) Gr dg2
p(G, B1,g2)
= -/(Brly A = a, G B 82] - By Y A a. B, g2)) p(p dga
Ec, Ey Y A = a, G1, B1, Gzl
EG, Ey Y A B G2
46

---

Substituting this expression back into the original conditional expectation, we obtain:
Eplva W G1, B1, G2j
G1, B1]
EG,
Ey A = G1. B1. G2]
EG.
Ey Y A B1, G2]
EG.
Ey [Y|
A a. B1. G2l
EG. Ey A = B1, G2
0
This establishes that the function (A, G1. B1. Gg) satisfies the first orthogonality condition in Condition C.2, [ G1, G2) Bi] 0.
To verify that it satisfies the second orthogonality condition, we proceed similarly by conditioning on G2:
B,, Ep No (Y, A,, G2; P) (Y, B,, G2; G2]
= =Ep [A( p G2) p(B))
EY A= a. G. B1. G Ey A= a, B1, G2]
p(A = B1, G2)
+ Ec.
Ey Y A = a, Bi. G2]
- EG, Eyly A= G,, G2
p B1) (G2)
Ep
|rry A= GI, Bi, G2| Ey A= a, Bi, G2]
p(G,, B1, G2)
Ec,
Ey Y A = a, B1, G2]
EG, Ey Y A= a, G.. Gr, Gz
G2
In the second equality, we integrate over A conditional on G2. We now rewrite the first term as an integral over (G1, B1) with respect P(G,, B, G2), and then simplify.
p (G,. B1) PG
Ep
Ey|Y A= a, G1, B1, Go
Y A = a, Bi, Gel
PGG, B1, G2)
+ EG. Byly A = a, B1, G2] - EEG, Ey Y = a, G. Bi, G2l G2
= / p(8,, by) (G2)
(Ey Y A = a,gi, b G2] Ry Y A a, b,, G2)) by G2) db, dg
p(g1, b1, G2)
Ep RGz Ey Y A = B1, Gg EG. Ey Y A = a, G B G2
G2
/ p(b..g)) g1) YY A 0,g1, bj Gzl Ey YY A a, bj. Gzl) dbi dgi
+ Epp EG. Ey Y = a. Bi. Gzl
EG. Ey Y A= a= a.
Br Go
G2
47

---

Next, we cancel the two components of the expression by invoking the conditional independence assumption C.4.
/ p(g), bi) (Eyly A a,gı, b1, -yl A = aı. G2l) dbj dgi
+ +Ep EG2 Ey A a a1, G2]
EG2 Er A = a, G1, BI, Go]
G2
= - (Br|| | A a, b1, Gb] - Ey Y A = a, b1, P(g) bI. a, G2)p(bi) dgi db,
Ep BG2 Ey [Y| A a, B1, Gol
EG,
Ey |Y A= = G. Go
G2
= -Ep EG.
By A= a, B1, G2]
EG. By Y A= G1. A G2l
G2
The second equality holds because we have the conditional independence assumption C.4, so: p(gı, b1) p(gi b1) p(b))
b,, A a. G2)p(bi), which justifies replacing the joint density in the integral. Furthermore, by the law of iterated expectations,
p(g
f B a, b1. G2] p(gi b a, G2) dg.
| A= a, b1, Gz). As result, the two integrals cancel out, yielding the final equality. To simplify this expression, compute both terms as iterated integrals.
Ep EG.
EY | AA a, B1, Gzl Go]
Ec, Byly | A=
= a, G1, B1. G2l
G2
=Iʃʃ v. (ply A == a., g2)) dy (B2) dga p(b] G2) db,
ppy A = a,, g2)] dy ((2) dg2 p(g, by G2) dg.
Based on Condition C.4, can simplify the second expression through the following derivation. First, we apply Condition C.4 (p(g) b,) = p(gı A = a, b G2)) to expand the conditioning set:
//fff y. (ply a, g1, b ga)) dy p(g2) dga p(gi, b G2) dg:
= =|Il/ y p(y A = a. g1, by dyp dg p(gi by)dgı p(b, G2)db1
-[]]] y ply = a. gi, by g2) dyp(gz)dg dgz p(gı A a, b g2 dg) p(b) G2)db1
Next, we express the integrand as a joint probability distribution over y and gz:
y ply.g. gi A = a, b,, 82) dydg; p(g2) dgg p(b, G2)db;
Using the chain rule of probability, we factorize the joint probability:
y p(g: y, A a. bi, b2) A = a, b b1. dg p(b, G2)db;
Finally, we marginalize over gı by integrating it out (since J p(gi A = a, b, B2) dg. 1), yielding the simplified expression:
= -Ilʃ y p(ul A a- bi, bi. dy p(b) G2)db;
48

---

This shows that the second term equals the first one. Thus, we conclude:
Ep
EG.
By Y A a, G2] EG, By Y A A= B.. Go
So far we have shown that the function g(A, G1, B1, G2) satisfies Condition C.5, so we obtain the orthogonality:
Ep (Y, (Y,A, B1, P2; (4% (Y, A, B1, G2; P) (Y A, G1, B1, P))] = 0.
Therefore, conclude that the variance decomposes
oa G.) (P) varp [te (Y, A, B1, G2; P)]
= varp ( (Y, A, G1, B1, G2; P)] + varp I (Y, - the A, G1. G2;
P))
2 o. G.)
Now extend this to the vectorized WCDE. Define: c (Ca)aEA Q = (Qa)aed where
Q. v/ AY G2; P) - 2/ (Y, A, G1, B1, BI, G2; G2;
For any Z, define (Y, A, Z; P) = (va A, Z; P)).E. Then, writing EnEA CY A, Z: P)
T (Y, A, Z: P) We notice that:
Ep[Q
G1, B1] = 0,
Ep[Q G2] = 0,
Then we obtain the orthogonality:
Ep(Q (YY, E A, G1, B1, G2; P)] =
Therefore, we conclude that the variance decomposes as:
i.B.G.) (P) varp [c U,YYA,B, G2; P)]
varp [c AYY,A, G1, B1, G2; P)] + varp [c T Q] =o oG + + varp(Q)e
B.)
This completes the proof.
C.4 Proof of Lemma 4.7
Proof. (Proof Lemma 4.7: Deletion of overadjustmenn mediator variables)
We begin by showing that the set (G), G2) remains a valid adjustment set (VAS), We can verify this
by comparing the identified WCDE functionals. For any adjustment set Z (Z,, Z2), the WCDE identified as
WCDE2 Ez, Ez, B(Y A Zı, 2z) - Ez, E(Y A = a- Zi. 2
For (Gi, B1, Gz). consider the a-term:
(2 B1, Gz) = Eg, B E(Y A G B1, Bz)
From the conditional independence
Y 1Lg Bj G1, G2,
(C.6)
it follows that
Y 4= a, G1 Bi G2 = Y = a, G1, G2].
Substituting this gives
Ta (G, B1. Gz) = EG, B Eg, E(Y E(Y
- = G. Gz)
Ec, Ec. [E(Y [TY
A = G.. G2)] = T. (G. G2).
49

---

and the same equality holds for a'. Therefore
B. = WCDE
By the uniqueness of the identified WCDE functional, any adjustment that yields the same functional as VAS must itself be VAS. Hence (G1, G2) is a VAS.
We compare the asymptotic variances of two VASs, (G1, G2) and (G1, Bi, G2). For elarity, we first focus on the first component of the WCDE in Equation (2.2), and let o o. denote its asymptotic variance. The extension to the full WCDE straightforward.
We begin by applying the law of total variance to decompose the variance under the larger adjustment set (G1, B1, G2):
Varl (Y, A, G1. B1. G2; P)] = Var (Y, A, G B1, P2; A, G1. G2]] + E [ar & (Y, B1, G2; P) A, G1. Gzll Bı,
Next, will show that under the conditional independence assumption C.6, the IF based on the adjustment set (G1, B1, G2) satisfies
Eplva (Y, A, BT G2; P) A.Y. G.] = V. (Y. A, G2, G2;
This equality allows relate the variances of the IFss via the law of total variance. We now compute the conditional expectation of tha (Y,Y, A, G1, B1, G2; P):
Ep [vt A.G. G1, B,, G2; P) A,Y, G2]
((G2) p( PG,,, B))
[A)) - Ey Y = a, G1, G2]} Ep
a= G2
p(A = a. G1, Bi, G2)
first
+ Ep Ea Ey Y A G1, Bi, G2]
G1. Bi]
(ii
+ Ep [G,,B, B, [BrlY A a, G1, Bi, G2]] G2 (P)
(ùi) third term
This equality holds because, given the conditional independence relation C.6, it implies the key simplification:
Ey A= a, B, Ey =A G₁, G2).
For the first conditional expectation term, we perform the detailed computation:
(G B1) (G2)
Ep
A a=
p(A, G1, Bi, G2)
G Bi) pl PG2)
= Ep
A a, G1 G2
[B(B]
A A= a, G1, G2)
p(A a, G1, G.)
GG,, ((G))
Ep
A A a. Y. G. G2
p(A G1, G2)
(BB A = G1, G2)
PGG, (()))(
EP
A= a. Y.G.Y
p(A G1, G2)
[p(B, A == a, G1, G.)
p[G1, pp))))
-P(b) A = a, G,, G2) db
M(A = G G2)
p(b A = a, G1, G2)
p(b)
A = 0, Y, Y,G G2 db
p(G bi) p(2)) p(b A a, G G2)
p(A= a, G1, G2)
bb
In the last line above, we expand the conditional expectation as an integral over B Now, under the conditional independence assumption C.6. it follows that
p(bi A a, Y, G1, G2) p(bi A a, G1, G2).
50

---

Substituting this into the integrand, the ratio simplifies to 1:
pb, A a, G1, G2)
p(G2) D(G
dbi
p(A = G1, G2)
bi
p(b
A = G1, G2)
p(G2) (G,,
db, db
p(1 G1. G2)
bb
p(b
G1) (GG p G2) db,
p(A G1. G2)
bb
p(G,)) (
p(bi
G1) db
p(A G1. G2)
/b
GG) (G2)
p(A G1: G2)
Since (Y A = a, G1, B1, G2) By(y A = a, G1 G2), so the second and third terms
simplify as follows:
Ep Er Er Y
a, G., BI, G1, B
=Ep FG. EG Y A G1 G2]l G
= -Eg Er = a, G Ga]
Ep [y Y A= a, G1, B1, G2] G2
-Ep EB, EEy Y A = G G2] G2
EG, [ A= 0 G1, G2
Combining these results, we obtain:
Er [Y, G, BI. G2; A,Y, G2]
Ta(A) p)( p(G))
Y Ey A = G1, G2)
G2
p(A a, G1, G2)
+ EG. E A a. G1, G2] + EEG Y A= a G2] 2Ta
= (Y, A, G1, G2; P)
That is, the conditional expectation of the influence function under the larger adjustment set equals the influence function under the smaller set.
Finally, applying the law of total variance, we establish the variance dominance relationship:
2 (.. varr (sba (Y, A, G,, B1, G2; P)] varr (Y, (Y, G1 G2; + Ep [varr (Y, A, G, B,, G2; A,Y, G1, Gz]]
2 2 (.)
The inequality follows because the conditional variance term is non-negative.
Now for the full WCDE estimator, define: с == (ca)aE for any Z, define + A, = A, (va (Y, Z; P))EA Then, writing ΣaEd (Y, = T (Y. A,
note that:
Ep [ (Y, BI, G2 A.Y, |,,, G3] (Y A, G1, G2;
51

---

So again by variance decomposition:
iG (G B. (.) (P) varp [c' (Y, G1. B1. G2; P)]
varp|Ep [c (Y, G1, B,, G2: P) A,Y, G1, G2] + Ep [vrrp [c (0,,,, A, G1, Bj, G2; P) A,Y, A.Y.G, G1, G2]]
varp c (Y, A, G1, G2; P)] + + Ep |aar (Y A, Gr, Bi, P2; A, G1, Ga]le
(P) + Ep varp (Y. A, Gr, B1, G2; P) Y, G1. G2]] 2 (.. This completes the proof.
G,),
C.5 Proof of Theorem 4.3
Proof. (Proof of Theorem 4.3)
Let Z = (Zz. Z2) be a valid adjustment set, where Z₁ = n M contains mediating variables and Ze Z Zi contains con-mediators (typically backdoor covariates).
We aim to construct an adjustment set that minimizes the asymptotic variance of the WCDE estimator by augmenting and pruning Z and Z2 based on conditional independencies in the DAG.
We prove the result through sequence of variance-reducing steps.
Step 1: Augmenting Zu with Non-media Parents of
According to Condition C1 and Condition C2, all backdoor paths for {A,Y) and {Z,, Y) are blocked by Z2, have the conditional independence:
(XiePa(Y) U (agr((Y)
(2) MLg A, Zz.
Then by Lemma 4.4, augmenting Z2 with X IEEP(Y) and X (EP((Y) reduces the variance:
o.z X (P) S o oz..
Step 2: Augmenting Z with Mediating Parents of Y
According to Condition C3, {M') \ Z1} b1g (Pa(Y) \ M' Z1, so we have the conditional independence:
(X)) Z1) dg A, Ze U X XIEP(()) U X4ER(()) Z₁. Then by Lemma 4.5, adding X3EPH(Y) further reduces variance:
oz (Z. Z2UX LEUXX (P) S o. Za UX
Step 3: Pruning Redundant Backdoor Variables from Z₂
Since XIgPa(Y) U UX4EPa(Y). Z₁ XaεPm(Y), and A together form the full set of parents of Y. according to the local Markov property, Y is conditionally independent of all non-descendant non-parent variables given its parents. Therefore, have the following conditional independence:
Y 1Lg Ze (XIePa(Y) (4EP)(Y)) XIEPa(Y) U Xag)a(Y) Z₁ X)E
Then by Lemma 4.6, removing extraneous elements of Z2 does not increase variance:
o (Z UXierry) (P) < (ZZ Z,XX, (P).
Step 4: Pruning Redundant Mediating Variables from Z
Note that X (I) UX4ePa(Y) together with Z, U X3εPa(Y) and A, constitute the full set of parents of Y. According to the local Markov property, Y is conditionally independent of all non-descendant non-parent variables given its parents, have the conditional independence:
Y M. Z Xzepu(r) XIePa(Y) U U4 XagPa(Y) A. Then by Lemma 4.7, variables Z₁ outside XaEPa(Y) can be excluded:
a (X XXeerr (P) < oz (Z X XX
52

---

Conclusion
Through these steps, we reduce the adjustment set to the minimal form that preserves validity and achieves minimal asymptotic variance. The final optimal adjustment set is:
XIePa(Y) U UX3ePa(Y) X4gPa(Y)
D Experimental Details
D.1 Synthetic Experiments Data-Cenerating Process
Edge coefficients. For each DAG edge, coefficients are sampled independently: with 50% probability from Uniform[-133 -0.5], otherwise from Uniform[0.5, 1.5], ensuring non-negligible effects.
Structural equations. Each variable X; the DAG generated recursively according a structural equation model:
X,
Σ jEPa(S)
Wزs (X,) Ess
where Whe is the sampled edge coefficient from parent j to i. The function fj(+) is applied in- dependently to each parent variable X, before combination and is selected uniformly at random from:
Identity: fxx 2, Sine: f(x sin(x), Cosine: f(x) cos(x).
Noise. The additive noise term E; is sampled independently from (O0 1/4) for each node with parents. Root nodes (i.e., nodes with no parents) are sampled from (O, 1).
Outcome transformation. 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.
Replications. We repeat this process for 50 randomly generated coefficient sets to assess the robustness of WCDE estimation. For each fixed DAG, adjustment set, and sample size, we additionally perform Monte Carlo experiment with size 100 estimate the empirical variance of the estimator across independent replications.
D.2 MILDEW Network
Table D.2: Variance and MSE of WCDE estimates on the MILDEW network (5 representative
VASs) under different sample sizes, Variable names are abbreviated as follows: mi meldug 3, m2:
middel_3, k: mikro_3. h2: mikro 2, 4: 1,i_1, l2: lai_2 2, 1z: lai_3.
250
500
1000
4000
10000
Adj. Set
Var
MSE
Var
MSE
Var
MSE
Ver
MSE
Var
MSE
Imn, m..
0.00666
0.00470
0.00269
0.00269
0.00104
0.00106
0.00020
0.00021
0.000100
0.00011
Uz. fa. mes, mi. ml
0.00512
0.005133
0.00301
0.00302
0.00114
0.00114
0.00027
0.00028
0,00012
0.00012
Hi. my.
0.00629
Imy.kl kl
0.014455
0.01521
0.00810
0.00810
0.00451
0.00455
0.00084
0.00084
0.00042
0.00042
0.01292
0.01400
0.00873
0.00873
0.00450
0.00460
0.00082
0.00083
0.00045
0.00046
0,01409
0.01527
0.00858
0.00888
0.00463
0.00472
0.90081
0.00081
0.00047
0.80047
The MILDEW network exhibits a highly complex structure with a large number of nodes and edges, resulting in enormous space of potential valid adjustment sets. To maintain both interpretability and representativeness, we adopted the following construction strategy. First, we identified all nodes that appear on the mediator paths between the treatment variable mikro_1 and the outcome variable meldug_4 4. We note that this particular treatment outcome pair admits no backdoor path on MILDEW network (potentially explaining the close performance of top performing VASs.) Next, we selected relevant nodes from the parents of both the treatment and outcome. We then systematically enumerated all possible combinations these nodes and filtered them using the adjustment criterion to ensure validity. This process yielded total of 122 valid adjustment sets.
53

---

Table D.3: Variance and MSE of WCDE estimates on the MILDEW network (top-50 VAS ranked by n = 10000). Abbreviations: & (mikro_ __), 12 (mikro_2) 4 (lai_ 1), 12 (lai_2) l3 (lai_3), mi
(meldug_33 3), m2 (middel_3) 3), m (mel Sheeldug_22 2).
250
500
1000
4000
10000
Adj. Set
Var
MSE
Var
MSE
Var
MSE
Ver
MSE
Var
MSE
Im,,, mag. k1
0,00466
0.00440
0,00269
0,00269
0.00104
0.00106
0.00020
0.00021
0.00010
0.00011 1
Ma. 121, may lo. M8
0,00465
0.00468
0.00275
0.00276
0.00125
0.00125
0.00026
0.00026
0.00011
0.90011
Ma. me,, ma. ks. A1
0,00470
0.00471
0.00270
0,00271
0.00123
0.00124
0,00025
0.00025
0.00011
0.00011 1
Uz, me1, meg.
0.00534
0.00534
0.00301
0,00302
0.00112
0.00112
0.00026
0.00027
0.00011
0.00012
Ma. mis ms.
0,00518
0.00520
0.00300
0,00301
0.00119
0.00119
0.00024
0.00024
0.00011
0.00012
Uo ma, mig. 1g. R]
0.00436
0.004366
0,00272
0.00272
0.00120
0.00120
0,00027
0.00027
0.00011
0.00012
Ima,, my. Az. A1
0.00443
0.00453
0.00244
0,00345
0.00117
0.00121
0,00023
0.00023
0.00012
0.00012
Pe. m. may, mog. A]
0.00616
0.00622
0.00319
0,00320
0.00121
0.00121
0.00026
0.00027
0.00012
0.00012
Us. ls my, ma.
0.00512
0.00513
0
0.00302
0.00114
0.001
0,00027
000002
0.00012
0.00012
Ps my m9. mog. A]
0,00578
0.00587
0.00299
0. 0.00299
0.001:1
0.00131
0,00027
0.00028
0.00012
0.00012
frn, mig. &]
0.00603
0.00668
0.00270
0 0.00271
0.00121
0.00121
0.00025
0.00027
0.00012
0.00013
Uz isi mi, mayy mag.
0,00555
0,00503
0,00307
00000
3.00134
0.00134
0,00088
000000
0.00012
3
Us m. may mag Ag.. A]
0.00476
0.00484
0.00285
0 0,00285
0.00143
0.00143
0.00026
0.00026
0.00012
0.00013
Ua. lai mis mag. Azz. R]
0,00446
0.00449
0,00272
0 00273
0.00133
0.00134
0,00227
0.00027
0,00012
O 0.00013
fro, my, ma, Ara A]
0.00531
0.00548
0.00269
0 1
0,00100
0.001
0,00227
0.00028
0.00012
0,00013
Ma, me, mas, mey, kg. k)
0.00469
0.004744
0.00282
O 000082
0.00132
0.00132
0.00025
0.00226
0,00013
0.00013
Uh,
0.00854
0,00588
0,00306
000000
0.00118
0.001
0,00000
000000
0.00013
Me. mes, ma. Aul
0,00518
0.00523
0.00325
0 000326
0.00155
0.00159
0
00000
0.00015
0,00013
Mo. la. may ma, Aal
0.005588
0.00552
0.00319
O 0.00321
0.001.22
0.00154
0.00030
0.00030
0.00013
3.00010
Ma. mes, mall
0.00534
0,00540
0.00303
40000
000015
0.00138
0,00030
0,00000
0.00013
O 010001
Me. las my, mal
0.00548
0.00550
0.00322
O 2200300
0.00147
0.00149
0
0.00032
0.00013
0,00010
Uz.
0.00524
0.00525
0,00330
O 000330
0000148
0,00151
0.00031
0000511
0,00014
0.00014
Us. my. ma. kal
0,00558
0.00566
0.00302
0,0005
0000142
0.00145
0,00029
0.00029
0.00014
0,00014
th. mi, mis ks. kll
0.00557
0.00538
0.00272
O 0,00275
100012
0.00131
0.00029
0.0022
0.00014
0,00014
Ui m. myy meg. lg. R]
0.00558
0.00559
0,00275
002016
28
0.001 12
0,00029
1500000
0.00014
0.00015
Ma. m. mi, mal
0,00693
0.00705
0.00335
0,00335
000000
0.00165
0,00033
000053
0.00015
0.00015
Uh m.
k]
0.00640
0.00640
0,00314
O 0,00011
011010
0.001 17
1
0,00015
0.00016
Mz, m. mas me]
0.00655
0.00667
0.00342
O 000343
000163
0.00163
0,00031
000002
0.00015
0.00015
Pa ls m. may mg]
0.00630
0.00642
0,00318
0,00319
0.00006
0.00166
0
0.00022
0,00015
0.00016
Us.
Ka]
0.00550
0.00556
0,00309
O 000300
000162
0.00163
0,00033
000034
0,00015
0.00016
Ua ma, may meg. Aal
0,00528
0.00543
0.00329
0 0,00229
0.00009
0.00109
0
000000
0.00016
0.00016
Us. ls. m. mns, mna, kal)
0.00526
0.00538
0.00309
0 0000000
000003
0.00163
0
000002
0.00016
0
Us la mi may mas
0.006488
00069
0,00385
0 00088
8000
0.00190
0,00441
000000
0,00018
0.00018
Mi la m. mas mas Aos. k]
0,00585
0.00585
0.00332
0 0,00332
0,00203
0.00207
0,00000
000011
0.00018
0.00018
Un la mis mis fog. 43]
0,00884
0. 008888
0,00423
0 2500425
000267
0,00275
0 0.00055
000006
0,00030
O 0.00020
Ui las may ma.
0.00767
0.00767
0.00367
00066
0000183
0.00190
0.00038
000030
0.00020
0.00020
Us la myy may. lo.. k]]
0.00694
0.00694
0.00335
O 0,00335
0001887
0.00194
0.00038
0.00099
O 0.00020 0.00020
Ui. lo my,
0.019555
0. 0.00955
0.00634
O 000000
0001558
0.00267
0.00052
000053
000053000000
0,000:0
0.00020
Uy la la. mas mus ky. R]]
0,00867
0.00870
0.004177
0 0,00128
0002243
0.00255
0.00057
0000557
0.00021
0.0002
Hy lo ly mys my. k]
0.01021
0.01021
0.00593
O 0,00597
0001251
0.00265
0 0.00054
0.00054
0.00023
O 0,00023
Un la m. mis m.. kall
0.00765
0.00766
0.00447
O 000048
0.00249
0.00053
000004
0.00024
0.00025
Hi ly my, my. Ra]
0,00749
0.00751
0.00425
0,00425
0O02273
0.00275
0.00046
0,00046
0.00024
O. 0.00024
U1. is. mas mal
0,00940
0.00940
0.00454
0 0,00455
000330
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54


Reviews:
[{"review_id": "A3Ds00GNzm", "reviewer": "Reviewer_tnMD", "summary": "The paper considers the estimation of \"Weighted Controlled Direct Effect (WCDE)\". CDE measures the effect of a treatment holding a mediator to a fixed level. WCDE averages over mediator distribution. The paper provides necessary and sufficient conditions for valid adjustment sets for unique identifiability of WCDE (condition 2.3), gives influence function for WCDE (Theorem 3.4), and shows that the optimal adjustment set that minimizes asymptotic variance is different from the ATE adjustment set.", "questions": "Please explain unique identifiability. \n\nPlease also explain limitations of the assumptions and method in detail. \n\nSatisfactory explanations to these points might increase my score to Accept.", "limitations": "Not adequate. The Q/A at the end says that they discuss future work, but I only find a summarization of the contributions and not much future work or limitation. Following the solicitation a clear specification and discussion of limitations and future work will be useful. For instance, the methods seem to be fully dependent on the correctness of causal DAG and correct specification of mediator variables, which may not hold in practice as is largely understood. Studying robustness when the causal DAG is not correct will be a future work.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "** Strength: **\n\nS1. The paper is in general well written. \n\nS2. I have not checked the proofs in the appendix, but proof sketches are provided in the main paper. One contribution is condition 2.3 (4), which uses interesting ideas for sufficiency and necessity. An example 2.5 is provided to illustrate the condition. \n\nS3. Similarly the proof idea of finding the optimal valid adjustment set is interesting and a proof sketch for 4 steps is given. I have not checked the proof in the appendix.  \n\n** Weakness: **\n\nPresentation: In some places a better clarity would be helpful.\n\n(a) in the introduction, better explanation can be provided for \"unique identifiability\" and the optimal adjustment set that minimizes variance, suggesting that there are multiple adjustment sets. \n\n(b) In condition 2.3, line 2.3, M should be defined since A and Y are not defined as treatment and outcome but arbitrary disjoint nodes. \n\n(c) In condition 2.3, backdoor and mediator paths are used for the first time. They should be defined for the sake of completeness. \n\n\nMinor typo in the 3rd condition in 2.3 -- \"are block\"", "was_revised": true, "final_justification": "The rebuttal addressed my concerns and I am increasing the score by 1 point.", "rebuttal": "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.\n\n**Weakness 1/Question 1: Clarify \"unique identifiability\" and the existence of multiple adjustment sets.**\n\nWe 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.\n\n\n**Weakness 2: Define $M$, and clarify the role of $A$ and $Y$ in Condition 2.3.**\n\nWe 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.\n\n**Weakness 3: Define \"backdoor\" and \"mediator\" paths.**\n\nWe 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:\n\n– **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 $Y$.\n\n– **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 $Z$.\n\n– **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 $A \\rightarrow M \\rightarrow Y$.\n\nThese 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.\n\n\n**Question 2: Discuss limitations.**\n\nWe 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.\n\n**Additional Experimental Results**\n\nFurther, to validate the robustness of our procedure under finite samples, we conducted additional experiments and will add them to our revised paper. \nWe construct two 7-node DAGs with multiple causal pathways, corresponding to the scenarios in Lemmas 1–4 (Figures 3 and 4).\nOur data is generated as follows:\n  >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.\n  >Each variable $X_i$ in the DAG is generated recursively according to a structural equation model:\n  >$$\n  >  X_i = \\sum_{j \\in \\text{Pa}(i)} w_{ji} \\cdot f_j(X_j) + \\varepsilon_i,\n  >$$\n  >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:\n  >\n>– **Identity**: $f(x) = x$  \n>– **Squared**: $f(x) = x^2$  \n>– **Exponential**: $f(x) = \\exp(x)$\n\n> 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 $\\mathcal{N}(0, 1)$.\n\n>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.\n\n\n**Estimation Method.**\nWe implement an AIPW estimator with:\n\n– **Q-model**: linear regression with spline-transformed features (degree 4, 20 knots)  \n\n– **g-model**: logistic regression for the treatment model.\n\nThe detailed form of the AIPW is included in the response to Reviewer Nm1A (Question 3).\n\n**Evaluation Metrics**\n\nFor 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.\n\n\n\n**Table 1. Average Variance for Figure 3**\n| Adjustment Set | \\$n=100\\$ | \\$n=400\\$ | \\$n=1000\\$ | \\$n=4000\\$ |\n| -------------- | --------- | --------- | ---------- | ---------- |\n| $G_1$, $G_2$        | 0.0110    | 0.0018    | 0.0007     | 0.0002     |\n| $B_2$, $G_1$        | 0.0385    | 0.0076    | 0.0031     | 0.0010     |\n| $B_2$, $G_1$, $G_2$ | 0.0282    | 0.0031    | 0.0011     | 0.0003     |\n\n**Table 2. Average MSE for Figure 3**\n| Adjustment Set | \\$n=100\\$ | \\$n=400\\$ | \\$n=1000\\$ | \\$n=4000\\$ |\n| -------------- | --------- | --------- | ---------- | ---------- |\n| $G_1$, $G_2$        | 0.0112    | 0.0019    | 0.0007     | 0.0002     |\n| $B_2$, $G_1$        | 0.0391    | 0.0077    | 0.0031     | 0.0010     |\n| $B_2$, $G_1$, $G_2$ | 0.0286    | 0.0032    | 0.0011     | 0.0003     |\n\n\n**Table 3. Average Variance for Figure 4**\n| Adjustment Set | \\$n=100\\$ | \\$n=400\\$ | \\$n=1000\\$ | \\$n=4000\\$ |\n| -------------- | --------- | --------- | ---------- | ---------- |\n| $G_1$, $G_2$        | 0.0142    | 0.0018    | 0.0007     | 0.0002     |\n| $B_1$, $G_2$        | 0.0615    | 0.0091    | 0.0048     | 0.0014     |\n| $B_1$, $G_1$, $G_2$ | 0.0496    | 0.0032    | 0.0012     | 0.0003     |\n\n\n**Table 4. Average MSE for Figure 4**\n| Adjustment Set | \\$n=100\\$ | \\$n=400\\$ | \\$n=1000\\$ | \\$n=4000\\$ |\n| -------------- | --------- | --------- | ---------- | ---------- |\n| $G_1$, $G_2$        | 0.0146    | 0.0018    | 0.0007     | 0.0002     |\n| $B_1$, $G_2$        | 0.0629    | 0.0093    | 0.0049     | 0.0015     |\n| $B_1$, $G_1$, $G_2$ | 0.0505    | 0.0033    | 0.0012     | 0.0003     |\n\n\n\nOur 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.\n\n\n[1] J. Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, 2009."}, {"review_id": "PZ0jSUK4mA", "reviewer": "Reviewer_Nm1A", "summary": "This paper studies nonparametric estimation of the Weighted Controlled Direct Effect (WCDE), which is an extension of the standard controlled direct effect (CDE) that averages over the mediator distribution, providing a robust estimate when treatment effects vary across mediator levels.", "questions": "- Could the authors include simulation studies to demonstrate the practical advantages of  the proposed methods. \n- How robust are the results (identifiability, influence function, and optimal adjustment set) to misspecification or uncertainty in the underlying DAG? Could the authors discuss the implications if the true DAG is unknown and has to be learned from data?\n- Could the authors elaborate on how practitioners can leverage the derived influence function (Theorem 3.4) to build efficient RAL estimators for WCDE?", "limitations": "N/A", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n- This paper studies nonparametric estimation of the Weighted Controlled Direct Effect (WCDE), which is an interesting and important extension of the standard controlled direct effect (CDE).\n- This paper defines a Valid Adjustment Set and establishes that Condition 2.3 is both necessary and sufficient for identifying a unique WCDE (Lemma 2.4).\n- This paper presents the identification theory and formulas for WCDE via Valid Adjustment Set.\n- This paper proposes three fundamental advances for WCDE in observational studies:\n  - Establishing necessary and sufficient conditions for the unique identifiability of WCDE, clarifying when it diverges from CDE.\n  - Deriving the influence function for nonparametric estimation of WCDE, considering the class of regular and asymptotically linear estimators.\n  - Characterizing the optimal covariate adjustment set that minimizes asymptotic variance, demonstrating how mediator-confounder interactions introduce distinct requirements compared to average treatment effect estimation.\n\n**Weaknesses:**\n\n- While the paper provides a strong theoretical foundation, the practical implementation of determining the \"O-set\" (optimal adjustment set) in real-world scenarios might be complex.\nThe entire framework, including the definition of VASs and the optimal adjustment set, assumes a known - - Directed Acyclic Graph (DAG) encoding the underlying causal structure. In many real-world applications, the true DAG is unknown and must be learned from data.\n- The paper's notion of \"optimal\" is based solely on minimizing asymptotic variance. While this is a critical statistical property, other practical considerations, such as bias in finite samples, robustness to model misspecification (beyond what's implied by RAL estimators), or computational feasibility for very high-dimensional data, are not discussed in detail.\n- The absence of experimental validation limits the immediate practical applicability and empirical understanding of the theoretical advancements presented.", "was_revised": true, "final_justification": "The authors effectively addressed my concerns regarding the practical implementation of determining the \"O-set\", and the need for experimental validation. They provided a clear discussion on causal discovery, new simulation results, and a practical explanation of the influence function's use in constructing efficient estimators. All of my concerns have been well-addressed. Thus, I raise my score by one point.", "rebuttal": "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.\n\n**Weakness 1: Practical feasibility of determining the O-set.**\nWe 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:\n\n>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.\n>\n>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.\n\n\n\n**Weakness 2: Other criteria beyond asymptotic variance.**\n\nWe 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.\n\n**Weakness 3/Question 1: Experimental Validation.**\n\nWe 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.\n\nOur 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.\n\n**Question 2: Robustness of results to uncertainty associated with DAG**\n\nIn 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.\n\nRegarding 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.\n\n    \n**Question 3: Use of the influence function in practice.**\n\nWe thank the reviewer for this important question. Indeed, including an exemplary efficient RAL estimator for WCDE would significantly strengthen our paper.\n\nIn 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.\n\nWe 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.\n\n[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.\n\n[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.\n\n\n[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.\n\n[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.\n\n[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.\n    \n[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.\n\n[7] Hoyer, Patrik, et al. \"Nonlinear causal discovery with additive noise models.\" Advances in Neural Information Processing Systems, 21, 2008.\n\n[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."}, {"review_id": "OLbfnhxsdr", "reviewer": "Reviewer_Hn2a", "summary": "This paper presents both necessary and sufficient conditions under which weighted controlled direct effect are identifiable. These necessary and sufficient conditions are proved to yield unique weighted controlled direct effect. Later, the paper also proposes a way to identify and verify optimal adjustment set that yields asymptotically lower variance of the estimated weighted controlled direct effect.", "questions": "See the weaknesses section above and provide your responses for those comments.", "limitations": "Limitations are not explicitly presented but future directions are discussed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is written well and easy to understand.\n2. This paper nicely extends the theoretical results from literature on effect identifiability and optimal adjustment set identification to the case of weighted controlled direct effect which has applications in domains such as fairness.\n\nWeaknesses:\n\n1. A major limitation of this paper is lack of experimental results. While the theoretical results are interesting and useful, experimental results verifying the theoretical results further significantly improve the paper.\n2. Considering the complexity of the paper, it benefits more from several illustrative figures explaining various aspects of the paper. For instance, more causal diagram based examples are needed to understand condition 2.3 (4).\n3. Minor: In L133, the phrase \"let $M'\\subseteq M$ be parents of $Y$\" needs a little refinement. Currently it reads like $M'=Pa(Y)$. Similarly in L145, clearly explain what counterfactual independence conditions are.", "was_revised": true, "final_justification": "All my questions are addressed during the discussion with the authors. I encourage authors to include the results on semi-synthetic data in the revised version.", "rebuttal": "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.\n\n**Weakness 1: Experimental results.**\nWe thank the reviewer for the excellent suggestions.\nIn 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.\n\n**Weakness 2: More illustrative figures.**\n\nWe 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.\n\nSimilarly, 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.\n\n**Weakness 3: Refinements in L133 and L145.**\n\nWe 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.\n\nAdditionally, 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. \nLet $Y_{a, \\mathbf{m}}$ 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:\n\n(1) **No unmeasured confounding of the treatment–outcome relationship**:\n$$\nY_{a, \\\\mathbf{m}} \\\\perp\\\\!\\\\!\\\\!\\\\perp A \\\\mid \\\\mathbf{Z}_2\n$$\n\n(2) **No unmeasured confounding of the mediator–outcome relationship**:\n$$\nY_{a, \\\\mathbf{m}} \\\\perp\\\\!\\\\!\\\\!\\\\perp \\\\mathbf{M} \\\\mid A, \\\\mathbf{Z}_2\n$$\n\nHere, $\\mathbf{Z_2}$ 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."}, {"review_id": "OrE7XQfNKc", "reviewer": "Reviewer_UNcX", "summary": "The paper makes three substantial contributions: conditions for identifiability, derivation of the influence function, and characterization of optimal adjustment sets for WCDE. These are theoretically significant and extend existing work on average treatment effect (ATE) adjustment.", "questions": "n/a", "limitations": "Since fairness is a key motivation, consider sketching a concrete fairness scenario where the WCDE provides insights unattainable through the CDE or ATE. Including a real-world case would make the example even more compelling.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nThe introduction effectively motivates the Weighted Controlled Direct Effect (WCDE) as a generalization of the Controlled Direct Effect (CDE), with clear links to fairness analysis and policy relevance.\n\nThe formal criteria for Valid Adjustment Sets (VAS) are well-defined and justified.\n\nThe contrast between CDE and WCDE is articulated well, especially regarding the limitations of fixing mediator levels arbitrarily.\n\nWeaknesses\n\nEven though the paper is theoretical, a small simulation study could help validate the efficiency gains from the optimal VAS, enhancing accessibility for applied researchers.\n\nThe identifiability and efficiency results hinge on having the true DAG and the absence of unmeasured confounding. Without access to the true DAG, the proposed method may no longer be valid.", "was_revised": true, "final_justification": "Indeed, as I mentioned in my previous comments, it is a good paper. I’m glad to see it accepted at NeurIPS.", "rebuttal": "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. \n\n**Weakness 1: Simulation to validate efficiency gains.**\n\nWe 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.\n\nOur 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.\n\n\n**Weakness 2: Discussion of DAG misspecification.**\n\nThank 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).\n\n**Limitation: Real-world fairness case study.**\n\nWe 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.\n\n\n\n\n\n[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.\n\n[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.\n\n[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.\n\n\n[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.\n\n\n[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.\n\n[6] T. J. VanderWeele. Controlled Direct and Mediated Effects: Definition, Identification and Bounds. Scandinavian Journal of Statistics, 38(3):551--563, 2011.\n\n[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. \\emph{Multivariate Behavioral Research}, 55(5):763--785, 2020."}]

Author rebuttal:

Meta-review: The paper studies estimation of the weighted controlled direct effect (WCDE), which generalizes the controlled direct effect (CDE) and is relevant in fairness analysis. The paper defines the notion of a valid adjustment set (VAS) for the WCDE, which extends the definition of a VAS for the CDE. The paper establishes that the existence of a VAS is necessary and sufficient conditions for the WCDE to be identifiable, and gives an identification formula in terms of the VAS. The authors derive the influence function of this quantity and characterize the "optimal" adjustment set (in the familiar sense of asymptotic efficiency, i.e., minimum asymptotic variance).

**Strengths.** The reviewers agree that the WCDE is an interesting and important extension of the CDE (Reviewers Nm1A, Hn2a, UNcX) which is sufficiently motivated by links to fairness. The paper is a well-grounded generalization of existing theory and provides several extensions of familiar, important definitions and existing results (valid adjustment sets, identification, influence functions, optimality). The reviewers also largely agreed that the paper was generally well-written, though there were a few suggestions for improving clarity, e.g., correct placement of some definitions such as "backdoor" and "mediator" paths (Rev. tnMD) and the addition of some illustrative figures (Rev. Hn2a).

**Weaknesses.** A main weakness pointed out by all reviewers was the lack of experiments to validate the theoretical results. The authors added results on semi-synthetic data during the rebuttal period (in their response to Reviewer Hn2a), reporting the variance and MSE of their estimates on three networks from bnlearn. The experiments show that, when the number of samples is relatively large (400 or 1,000), the adjustment set selected by their method yields an estimator with lower variance and MSE than other adjustment sets. Another weakness pointed out by two reviewers (Reviewers Nm1A and UNcX) is that the results hinge on having the true DAG in the absence of unmeasured confounding. The authors acknowledge this point and mention that they will explicitly discuss this limitation in the revision. While this is indeed an important limitation, I don't think it is a major weakness of this work, since it provides the necessary first step for the development of methods which are robust to misspecification or which simultaneously consider both structure learning and effect estimation. A similar comment applies to another one of the limitations, mentioned by Reviewer Nm1A: that "optimality" is only based on asymptotic variance and not finite-sample bias.

**Conclusion.** The paper makes a solid technical contribution to an important area of causality. Even without the experiments and other improvements from the rebuttal, the paper is ready to be accepted.

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**Minor comment to the authors:** I believe the term "unique identifiability" is redundant. Saying that a quantity is "identifiable" already implies that it is unique. To avoid confusion, I would suggest removing the term "unique", unless you have something different in mind which I've missed.
Decision: Accept (poster)