Title: Decoding Causal Structure: End-to-End Mediation Pathways Inference
Venue: NeurIPS 2025
Accepted: yes
Keywords: Mediation Pathways Inference
Abstract: Causal mediation analysis is crucial for deconstructing complex mechanisms of action. However, in current mediation analysis, complex structures derived from causal discovery lack direct interpretation of mediation pathways, while traditional mediation analysis and effect estimation are limited by the reliance on pre-specified pathways, leading to a disconnection between structure discovery and causal mechanism understanding. Therefore, a unified framework integrating structure discovery, pathway identification, and effect estimation systematically quantifies mediation pathways under structural uncertainty, enabling automated identification and inference of mediation pathways. To this end, we propose Structure-Informed Guided Mediation Analysis (SIGMA), which guides automated mediation pathway identification through probabilistic causal structure discovery and uncertainty quantification, enabling end-to-end propagation of structural uncertainty from structure learning to effect estimation. Specifically, SIGMA employs differentiable Flow-Structural Equation Models to learn structural posteriors, generating diverse Directed Acyclic Graphs (DAGs) to quantify structural uncertainty. Based on these DAGs, we introduce the Path Stability Score to evaluate the marginal probability of pathways, identifying high-confidence mediation paths. For identified mediation pathways, we integrate Efficient Influence Functions with Bayesian model averaging to fuse within-structure estimation uncertainty and between-structure effect variation, propagating uncertainty to the final effect estimates. In synthetic data experiments, SIGMA achieves state-of-the-art performance in pathway identification accuracy and effect quantification precision under structures uncertainty, concurrent multiple pathways, and nonlinear scenarios. In real-world applications using Human Phenotype Project data, SIGMA identifies mediation effects of sleep quality on cardiovascular health through inflammatory and metabolic pathways, uncovering previously unspecified multiple mediation paths.

Decoding Causal Structure: End-to-End Mediation Pathways Inference

Abstract

Causal mediation analysis is crucial for deconstructing complex mechanisms of action. However, in current mediation analysis, complex structures derived from causal discovery lack direct interpretation of mediation pathways, while traditional mediation analysis and effect estimation are limited by the reliance on pre-specified pathways, leading to a disconnection between structure discovery and causal mechanism understanding. Therefore, a unified framework integrating structure discovery, pathway identification, and effect estimation systematically quantifies mediation pathways under structural uncertainty, enabling automated identification and inference of mediation pathways. To this end, we propose Structure-Informed Guided Mediation Analysis ( SIGMA ), which guides automated mediation pathway identification through probabilistic causal structure discovery and uncertainty quantification, enabling end-to-end propagation of structural uncertainty from structure learning to effect estimation. Specifically, SIGMA employs differentiable Flow-Structural Equation Models to learn structural posteriors, generating diverse Directed Acyclic Graphs (DAGs) to quantify structural uncertainty. Based on these DAGs, we introduce the Path Stability Score to evaluate the marginal probability of pathways, identifying high-confidence mediation paths. For identified mediation pathways, we integrate Efficient Influence Functions with Bayesian model averaging to fuse within-structure estimation uncertainty and between-structure effect variation, propagating uncertainty to the final effect estimates. In synthetic data experiments, SIGMA achieves state-of-the-art performance in pathway identification accuracy and effect quantification precision under structural uncertainty, concurrent multiple pathways, and nonlinear scenarios. In real-world applications using Human Phenotype Project data, SIGMA identifies mediation effects of sleep quality on cardiovascular health through inflammatory and metabolic pathways, uncovering previously unspecified multiple mediation paths.

1 Introduction

Causal Mediation Analysis (CMA) has emerged as a critical framework bridging causal inference and machine learning. Recent studies [47, 27] emphasize the evolution of CMA from identifying causal effects to elucidating underlying mechanisms, a transition essential for developing machine learning systems capable of intervention reasoning, cross-domain knowledge transfer, and interpretable explanations [10, 2, 52]. In particular, analyzing causal mediation pathways involving multiple mediators arranged in parallel, sequential, or networked structures [58, 9, 28] provides deeper mechanistic

insights in high-dimensional contexts commonly encountered in biology, social networks, and behavioral sciences [21, 26, 40]. For instance, elucidating how sleep quality impacts cardiovascular health in Human Phenotype Projects (HPP) [36] requires modeling its interplay with multiple mediators ( e.g., inflammation, metabolic) across interrelated pathways [30, 61, 7]. Therefore, quantifying the effects of mediation pathways under causal structural uncertainty remains a challenge.

Although analyzing complex mediation pathways is both theoretically and practically significant, current methods face several limitations: (i) Structure Discovery (SD): While mainstream causal discovery algorithms [56, 55, 72] can automatically learn causal structures, they lack principled mechanisms for quantifying and propagating structural uncertainty, undermining the reliability of downstream analyses. (ii) Pathway Identification (PI): Current structure discovery methods [1, 29] fail to explicitly identify specific mediation pathways. Meanwhile, classical mediation analysis approaches [3, 27] rely on pre-specified paths, which become infeasible in high-dimensional data due to the exponential growth in the number of candidate paths. (iii) Effect Estimation (EE): Semiparametric methods like Efficient Influence Function (EIF) [6] offer theoretical guarantees ( e.g., multiple robustness) [59, 71, 41] but typically assume that the mediation structure is known and correctly specified, which creates a contradiction with the structural uncertainty faced in the previous two stages. Consequently, a gap between current methods becomes evident in high-dimensional settings like HPP, where discovery yields causal structure without pathway interpretation, while traditional mediation is confined to predefined paths, missing underlying causal mechanisms. In this work, we argue that a unified framework integrating SD, PI, and EE can quantify the mediating pathways under structural uncertainty in real-world settings.

To this end, we propose an end-to-end S tructureI nformed G uided M ediation A nalysis (SIGMA) framework, which integrates probabilistic causal structure discovery with uncertainty quantification to guide automatic mediation pathway identification, and propagates structural uncertainty to robust effect estimation, thus enabling integrated inference of causal structures and mediation effects. Specifically, differentiable Flow-Structural Equation Models [64] are leveraged to learn edge parameters encoding structural uncertainty, from which an ensemble of directed acyclic graphs (DAGs) is subsequently sampled. Building on this DAG ensemble, we introduce the Path Stability Score which performs Bayesian inference by evaluating the marginal probability paths across DAGs to select high-confidence mediation pathways, thereby circumventing manual path pre-specification. For each identified mediation pathway, SIGMA employs Efficient Influence Functions to estimate pathway-specific mediation effects, using nuisance functions tailored to each DAG structure. Subsequently, these pathway-specific estimates are aggregated across the DAG ensemble via Bayesian model averaging (BMA) [48], propagating structural uncertainty into the final causal effect estimates. Furthermore, SIGMA integrates a variational autoencoder-based imputation module to explicitlymodel latent dependencies among variables, enabling accurate imputation of missing values. We empirically demonstrate SIGMA's effectiveness on high-dimensional, heterogeneous HPP data 1 , identifying and quantifying mediation pathways connecting sleep quality to cardiovascular health through inflammatory and metabolic mediators. The contributions are summarized as follows:

We propose an end-to-end framework, SIGMA, unifying probabilistic structure discovery with uncertainty quantification, high confidence path identification, and effect estimation incorporating structural uncertainty, to address mediation analysis under uncertainty structures.

We introduce the Path Stability Score to identify mediation by quantifying their stability across posterior directed acyclic graph samples, thus eliminating manual pre-specification.

We develop a structure-guided estimation approach, employing Efficient Influence Functions (EIF), combined with Bayesian model averaging, enabling propagation of structural uncertainty and inference by leveraging EIF properties.

Extensive experiments validate SIGMA on real-world HPP data, revealing complex and previously unspecified mediation pathways linking sleep quality to cardiovascular health through inflammatory and metabolic, demonstrating its interpretability in real-world.

2 Dataset

Human Phenotype Project (HPP) 1 . We conduct our real-world evaluation using the HPP dataset, a deeply phenotyped cohort comprising over 6,000 individuals with multi-night home sleep apnea

a Sleep Characteristics

C

·Restorative sleep ratio

·height

. Sleep duration

weight

·Sleep continuity

Appendicularleanmass

· Sleep efficiency

Visceralfatratio

· Snoring ratio

·A/Gfatratio

Snoring ratio

·OSA

CardiovascularSystem

· HRV (during night)

: Oxygen desaturation index

Vascular Health

Carotid ultrasound IMT

Carotid ultrasoundplaque

Blood pressures

Metabolic Pathways

.Pulserate

·GutMB-butyrate producers

Inflammation

·BMI variables

GutMB-propionateproducer

●sleep characteristics ·inflammation variables

·GutMBvitaminproducers

GutMBprobioticfermenters

.GutMB-lps producers

·GutMB-tma producers

●metabolicvariables·cardiovascularvariables

·GutMBakkermansia

GutMB-sulfate reducers

→Existing pathways==>Discovered pathways

GutMB shannon

·GutMB-proteolytic bacteria

M;→Y|A

M→Y|A

A=0.0

A=0.0

± 6 month from visit

A=1.0

A=1.0

14 days after visit

2 days

1

1

Measure Collected:

3-nights sleep

Stool sample

Blood tests from

M→Y|A

M

AMY

·Bloodpressure

monitoring

collection

Health Maintenance

·Blood sample

Organization (HMO)

·Body composition

14-day post-visit period:

Continuousmeasuresover the

·Anthropometrics

Carotidultrasound(IMT)

· Food logging

A=0.0

A=1.0

testing (HSAT) data across 16,000+ nights. Beyond sleep, HPP includes multimodal data covering 16 other body systems that represent a wide array of physiological functions and environmental exposures [34]. This coordinated collection process, as shown in Figure 1(b), provides temporally aligned, longitudinal measurements across multiple physiological systems. The integration of sleep monitoring, clinical biomarkers, imaging, and omics data within a ± 6-month window requires considerable logistical effort and sustained participant engagement. These characteristics make the HPP dataset a particularly valuable testbed for evaluating SIGMA's ability to recover complex mediation pathways under structural uncertainty. In total, we utilize data from 11 body systems, covering over 500 raw clinical and physiological attributes. To support tractable and clinically meaningful mediation analysis, we aggregated these into 62 domain-informed features by grouping related variables across systems. Each feature was designed to reflect a coherent physiological process suitable for pathway-level causal interpretation. Among these, five representative categories analyzed in our work: Sleep Characteristics , Body Mass Index (BMI) , Cardiovascular System , Inflammation and Metabolic Pathways , are illustrated in Figure 1(a). Detailed data descriptions can be found in the Appendix D. As depicted in Figure 1(c), SIGMA framework is capable of validating previously reported mediation pathways [19] while also uncovering novel, clinically meaningful causal mediation mechanisms spanning different systems. We provide results to demonstrate this in Section 5.3, where we quantify these pathways and assess their natural effect.

Synthetic Data. To evaluate SIGMA's ability to identify mediation pathways under controlled yet realistic conditions, we generate synthetic datasets that mirror key statistical and structural characteristics of real-world data (Figure 1(d)). Specifically, we simulate missing data patterns with both missing completely at random (MCAR) [50] and missing at random (MAR) mechanisms to reflect non-random measurement gaps common in clinical studies, and introduce heterogeneous variable types (continuous, binary, categorical) and nonlinear relationships to mimic biological complexity. Each synthetic dataset is constructed from a randomly sampled DAG using an Erd˝ os-Rényi (ER) [20] model, followed by structural equation modeling (SEM) [46] with configurable nonlinearity and noise. We explicitly embed mediation structures-parallel, chain, or hybrid-between a designated treatment and outcome, with tunable effect strengths to ensure identifiable direct and indirect effects.

Functional mechanisms include both linear and nonlinear mappings (e.g., polynomial, sinusoidal), and exogenous noise is drawn from mixed distributions to induce non-Gaussianity. Natural Direct/Indirect Effects (NDE/NIE) [45, 49] are estimated via Monte Carlo simulations to facilitate quantitative benchmarking. This synthetic setup enables systematic evaluation of SIGMA's robustness across varied structural regimes, pathway configurations, and data quality levels. See Appendix E.1 for detailed generation procedures and E.2 for embedded mediation pathway structures.

3 Causal Mediation: Definitions, Notation, and Identification

We consider the problem of causal mediation analysis in observational data. Let A denote the treatment variable, Y the outcome variable, M = { M 1 , . . . , M k } a set of k potential mediators, and C a set of covariates. The variables may be continuous, discrete, or mixed-type. Let V = { A,Y } ∪ M ∪ C be the full set of observed variables and assume the underlying causal structure is a DAG G = ( V, E ) , where Pa G ( V i ) denotes the parent set of node V i . We assume that the true graph G ∗ satisfies the Causal Markov and Faithfulness assumptions, which enable conditional independencies to be inferred from the graph. Appendix A.1 provides a detailed discussion of these assumptions, including Causal Sufficiency.

We adopt the Potential Outcomes Framework [50, 57] to define causal mediation effects. Let M ( a ) be the potential mediator value under treatment A = a , and Y ( a, m ) the potential outcome if A = a and M = m . We focus on two core mediation effects comparing treatment level a to reference level a ∗ : (1) The NDE [45] is defined as NDE ( a, a ∗ ) = E [ Y ( a, M ( a ∗ )) -Y ( a ∗ , M ( a ∗ ))] , quantifying the effect of changing A from a ∗ to a on Y while holding M fixed at its natural level under A = a ∗ . (2) The NIE [49] is defined as NIE ( a, a ∗ ) = E [ Y ( a, M ( a )) -Y ( a, M ( a ∗ ))] , capturing the effect on Y mediated through M when A change from a ∗ to a . When the true causal structure is unknown, our goal is to estimate these effects under structural uncertainty. Under certain assumptions [45], the Total Effect, TE ( a, a ∗ ) = E [ Y ( a ) -Y ( a ∗ )] can be decomposed as TE = NDE + NIE.

Identifying NDE and NIE from observational data requires several key assumptions [8, 45]. We use ⊥ ⊥ to denote conditional independence. Appendix A.3 discuss more about following assumptions, including strategies for practical consideration and the theoretical implications of violation. Throughout, identification denotes recovery of NDE and NIE under ignorability, consistency, and positivity; path identification denotes that a path attains high posterior support as quantified by the PSS. Estimation then proceeds via influence-function-based and plug-in estimators under these assumptions.

Consistency : if A = a ′ , then M = M ( a ′ ) , and if A = a ′ and M = m , then Y = Y ( a ′ , m ) , for all relevant treatment levels a ′ (e.g., a ′ ∈ { a, a ∗ } ) and all m in the support of M .

Ignorability : The following conditional independence relations hold almost surely for all relevant values a, a ∗ , m, m ′ : (i) A ⊥ ⊥ { Y ( a ∗ , m ) , M ( a ) } | C ; (ii) M ⊥ ⊥ Y ( a, m ′ ) | A = a, C ; and (iii) Y ( a ∗ , m ) ⊥ ⊥ M ( a ) | C .

Positivity : For all a and c in the support of A and C , the treatment propensity score π ( a | C ) ≡ Pr ( A = a | C = c ) ∈ ( c 1 , c 2 ) for some constants 0 < c 1 ≤ c 2 < 1 . The conditional density function for discrete M = m given A and C = c is bounded between [ ρ 1 , ρ 2 ] for some constants 0 < ρ 1 ≤ ρ 2 < ∞ , almost surely for all relevant a, c , and m .

These assumptions collectively allow identification of NDE and NIE from observational data, forming the foundation of our mediation analysis under unknown causal graph structure.

4 Methodology

4.1 SIGMA Framework Overview

We propose Structure-Informed Guided Mediation Analysis (SIGMA), a unified framework for causal mediation analysis under structural uncertainty. SIGMA operates in an end-to-end fashion, integrating causal discovery, pathway identification, and effect estimation within a probabilistic graphical modeling paradigm. Given observational data over treatment, outcome, mediators, and confounders, SIGMA performs probabilistic structure discovery via a flow-based structural equation model (FlowSEM), generating a posterior ensemble of DAGs that capture causal uncertainty (Section 4.2). It then identifies stable mediation pathways using a path stability score (PSS), which quantifies the marginal support for each path across sampled DAGs (Section 4.3). For each retained pathway, SIGMA estimates natural direct and indirect effects using structure-guided nuisance estimation and

a hybrid influence-function/plugin strategy, with final estimates aggregated via Bayesian model averaging to propagate structural uncertainty (Section 4.4). This unified design enables automated, uncertainty-aware mediation analysis in complex high-dimensional systems.

4.2 Probabilistic Structure Discovery

To model structural uncertainty in causal mediation, SIGMA employs a Flow-SEM to learn a distribution over DAGs from observational data. Let V = { V 1 , . . . , V p } denote the set of variables, the goal is to infer a posterior distribution P ( G | D ) over DAGs G = ( V, E ) that encode the conditional dependencies among these variables. The validity of this structure discovery step relies on several core assumptions, including the Causal Markov, Faithfulness, and Causal Sufficiency assumptions. Refer to Appendix A for a detailed discussion.

4.2.1 Flow-SEM Model Formulation

In Flow-SEM, each variable V i ∈ V is modeled as a stochastic function of its parents in a candidate DAG G , with the structural equation:

where Pa G ( V i ) denotes the parent set of V i , and f i , σ i are flexible neural network functions parameterized by ϕ i . The collection Φ = { ϕ i } p i =1 defines the global functional parameters. This formulation accommodates nonlinearities and heteroskedasticity in the data-generating process.

To enable differentiable structure learning, SIGMA introduces a continuous weight matrix W θ ∈ R p × p , where each entry [ W θ ] ji reflects the strength or existence of a directed edge from V j to V i . A zero-diagonal [ W θ ] ii = 0 is imposed to prevent self-loops. The parent set Pa G ( V i ) is inferred from the sparsity pattern of W θ . Under this, the joint distribution P ( V | W θ , Φ) can be derived via a change-of-variables formulation using normalizing flows. Given an observed data sample V ( j ) , its log-likelihood is computed as:

where u ( j ) i = ( V ( j ) i -f i ( Pa G ( V ( j ) i ); ϕ i )) /σ i ( Pa G ( V ( j ) i ); ϕ i ) represents the standardized noise for variable V ( j ) i . This formulation enables likelihood-based training of both structural and functional components via stochastic optimization.

4.2.2 Model Learning and Optimization

Model parameters ( W θ , Φ) are optimized by minimizing a regularized loss that balances data fit, structural sparsity, and acyclicity:

̸

where h ( W θ ) = tr (exp( W θ ⊙ W θ )) -p serves as a continuous relaxation for enforcing acyclicity, exp( · ) is the matrix exponential and ⊙ denotes element-wise multiplication; ∥ W θ ∥ 1 = ∑ i = j | [ W θ ] ji | encourages sparsity. Further technical details and acyclicity properties are discussed in Appendix A. Gradient-based optimization is used to train the model. Once trained, W θ captures both the functional and structural dependencies among variables.

4.2.3 Approximate Structural Posterior and DAG Sampling

To approximate the structural posterior P ( G | D ) , SIGMA interprets each entry in W θ as defining a marginal edge inclusion probability via:

where σ ( · ) is the sigmoid function. Candidate DAGs { G s } N DAG s =1 are generated by sampling each edge independently as Bernoulli ( p ij ) , followed by cycle removal to ensure graph validity.

This ensemble of sampled DAGs provides a Monte Carlo approximation of the posterior over causal structures, capturing both edge-level and pathway-level uncertainty. These samples are used downstream for pathway selection and mediation effect estimation (see Sections 4.3 and 4.4).

4.3 Automated Mediation Pathway Identification

Given a posterior ensemble of sampled DAGs { G s = ( V, E s ) } N DAG s =1 , SIGMA identifies mediation pathways that are robust to structural uncertainty by quantifying the posterior support for each candidate path. This is achieved through the Path Stability Score (PSS), which estimates the marginal probability of a directed path being present in the true causal structure.

Formally, let π = ( V 0 → V 1 → ··· → V k ) denote a directed acyclic path from treatment node V 0 = A to outcome node V k = Y . The PSS of path π is defined as:

where I ( π ⊆ G s ) = 1 if all directed edges in π are present in graph G s , and 0 otherwise. The term N valid denotes the number of acyclic graphs retained after cycle removal. As shown in Appendix B.1, PSS is an unbiased and consistent estimator of the posterior inclusion probability P ( π ⊆ G | D ) .

To construct the candidate set of mediation paths, SIGMA enumerates all simple directed paths from A to Y in each valid DAG G s , subject to a maximum length constraint for computational tractability. These paths are aggregated across DAGs, and their frequencies are normalized to obtain the empirical PSS scores. Paths with stability above a user-specified threshold τ ∈ [0 , 1] are retained as highconfidence mediation candidates, forming the set: Π stable = { π | PSS ( π ) ≥ τ } . This selection process effectively propagates structural uncertainty from edge-level variation to the pathway level, allowing downstream effect estimation to focus on statistically supported mediation mechanisms. A theoretical justification for using PSS as a Monte Carlo estimator and its asymptotic properties is provided in Appendix B.1 and B.2.

4.4 Mediation Effect Estimation

Once a set of stable mediation pathways Π stable is identified from the DAG ensemble, SIGMA proceeds to estimate path-specific NDE and NIE for each retained pathway π ∈ Π stable. Given that each pathway is supported by a subset of DAGs { G s | π ⊆ G s } , effect estimation must account for both within-structure statistical uncertainty and between-structure causal ambiguity. SIGMA achieves this by combining structure-specific nuisance estimation with hybrid estimators and Bayesian model averaging (BMA). Appendix A.3 details the assumptions required for valid estimation, including ignorability, consistency, and positivity conditions.

4.4.1 Structure-Guided Nuisance Estimation

For each stable pathway π ∈ Π stable, SIGMA estimates the nuisance functions ˆ η s by leveraging structure-specific information from each DAG G s in the ensemble. The conditioning sets for all components of ˆ η s are defined strictly by the parent sets Pa G s ( · ) within the corresponding DAG. Neural networks are used to fit the conditional relationships implied by these parent sets. To reduce estimation bias and enhance robustness, we use K-fold cross-fitting to compute out-of-sample predictions across the dataset. This procedure yields all necessary components for downstream effect estimation via both the EIF and plugin estimators, including the required counterfactual outcome predictions.

4.4.2 DAG Effect Estimation

After obtaining nuisance function estimates ˆ η s for a specific DAG G s (as detailed in Section 4.4.1), the SIGMA framework calculates path-specific NDE and NIE for a path π . These constitute the single-DAG effect vector ˆ θ s ( π ) = ( ̂ NDE s ( π ) , ̂ NIE s ( π ) ) . SIGMA employs a hybrid estimation strategy: for paths of length L ( π ) (number of nodes) equal to 3 ( i.e., A → M → Y ), an influence function based estimation method is utilized; for paths with L ( π ) > 3 , a plugin estimator is adopted. All effect quantifications are based on a comparison between a pre-specified exposure level a and a reference level a ∗ . For a binary exposure A , these levels are 1 and 0 , respectively. For a continuous exposure A , a ∗ is its estimated mean ˜ µ A and a is ˜ µ A + k · ˜ σ A (with k = 1 in this study), where ˜ µ A and ˜ σ A are estimated from the first K-Fold training set for the specific (path, DAG) combination.

Influence Function-based Estimation ( L ( π ) = 3 ) : The point estimate ˆ θ s ( π ) is the sample mean of the corresponding influence function scores ψ i ( Z i ; ˆ η s , G s ) , i.e., ˆ θ s ( π ) = 1 n ∑ n i =1 ψ i ( Z i ; ˆ η s , G s ) .

For a binary exposure A , the influence functions ψ NDE ( Z i ; ˆ η s ) and ψ NIE ( Z i ; ˆ η s ) are constructed within the SIGMA framework by combining cross-fitted estimates of several nuisance functions. These include the propensity score ˆ P ( A i = a | C A,i ) , outputs from the mediator model M M | A,C M , which characterizes the conditional behavior of M i given A i = a and covariates C M,i , and the conditional expectation of the outcome ˆ E [ Y i | A i = a, M i , C Y,i ] . These IFs are formulated to provide efficient, bias-corrected estimates for the target counterfactual expectations (e.g., E [ Y ( a ′ , M ( a ′′ ))] ) that comprise NDE and NIE. The specific mathematical expressions for these influence functions as utilized within SIGMA are detailed in Appendix A.3.4.

For a continuous exposure A , SIGMA constructs and applies a specific influence function. This function aims to correct plugin estimates of counterfactual expectations (e.g., γ aa ∗ = E [ Y ( a, M ( a ∗ )) ), which form the NDE and NIE, by integrating cross-fitted nuisance components using the data-driven intervention levels ( a, a ∗ ) . These key nuisance components include: (i) the conditional expectation of A , ˆ µ A ( C A,i ) ; (ii) an estimate of the global homoscedastic residual variance of A , ˆ σ 2 A | C A ,res ; (iii)

the conditional expectation of the outcome ˆ µ ( i ) Y | A = level ,M obs ,C Y ; and (iv) individual plugin predictions of the target counterfactual expectation ˆ γ aa ∗ ,i and their sample mean ˆ γ aa ∗ . The influence function score ψ γ aa ∗ ( Z i ) used in SIGMA to estimate a counterfactual expectation (such as γ aa ∗ ) is defined as:

The overall influence functions for NDE and NIE are derived from linear combinations of these component scores. The theoretical motivation and property analysis of this SIGMA-specific influence function are detailed in Appendix A.3.5. For all influence function-based estimates, the single-DAG variance ˆ V ( ˆ θ s ( π )) is estimated by 1 n 2 ∑ n i =1 ( ψ i -¯ ψ ) 2 , where ¯ ψ = ˆ θ s ( π ) .

Plugin Estimation ( L ( π ) > 3 ) : For paths longer than three nodes, SIGMA utilizes a plugin estimator. NDEandNIEare computed by substituting the relevant cross-fitted counterfactual outcomes (i.e., individualized ˆ µ ( i ) Y ( a, M ( a ′ )) predictions from Section 4.4.1) into their defining expressions, followed by averaging over the sample. Specifically, ̂ NIE s ( π ) = 1 n ∑ n i =1 ( ˆ µ ( i ) Y ( a, M ( a )) -ˆ µ ( i ) Y ( a, M ( a ∗ )) ) . Its variance ˆ V ( ˆ θ s ( π )) is calculated as 1 n ( 1 n -1 ∑ n i =1 (∆ ( i ) CF -∆ CF ) 2 ) , where ∆ ( i ) CF denotes the individuallevel counterfactual difference.

4.4.3 Aggregation and Uncertainty Propagation

To generate final mediation effect estimates that account for structural uncertainty, SIGMA aggregates single-DAG estimates ˆ θ s ( π ) and their variances ˆ V ( ˆ θ s ( π )) for a path π across the N path supporting DAGs. The Bayesian Model Averaged point estimate is ˆ θ BMA ( π ) = 1 N path ∑ N path s =1 ˆ θ s ( π ) , using uniform weights for each supporting DAG G s . The total BMA variance, ˆ V ( ˆ θ BMA ( π )) , combines the average within-DAG variance (reflecting estimation uncertainty for each structure) and the between-DAG variance (capturing structural uncertainty across different structures) via the law of total variance:

Here, ˆ V ( ˆ θ s ( π )) is the single-DAG variance estimate from Section 4.4.2. Confidence intervals are then constructed using the standard normal approximation: ˆ θ BMA ( π ) ± z 1 -α/ 2 √ ˆ V ( ˆ θ BMA ( π )) . This aggregation framework ensures that both estimation uncertainty (within-DAG) and structural uncertainty (between-DAG) are rigorously propagated into the final mediation effect estimates.

5 Experiments

We evaluate SIGMA across two key tasks: (1) causal structure discovery and (2) causal mediation analysis, covering both synthetic datasets with known ground-truth effects and a real-world medical cohort from the HPP. For structure discovery, we assess the accuracy of learned causal graphs in

various settings with different graph topologies, nonlinearity levels, and data types. For mediation analysis, we test SIGMA's ability to estimate direct and indirect effects of identified mediation pathways. The synthetic data allows controlled benchmarking, while the HPP cohort provides a complex, high-dimensional setting with clinically relevant mediation structures. A full list of results and details of the experimental setup are in Appendix F.1.

Experimental setup Our experimental evaluation comprises two primary tasks: (1) performance assessment of causal structure discovery, and (2) end-to-end causal mediation analysis evaluation. For synthetic data experiments, we construct 10 benchmark datasets that systematically vary feature dimensionality, degree of nonlinearity, and pathway structural complexity to assess SIGMA's robustness across different complexity regimes. Each dataset contains 6,000 samples with mixed variable types (60% continuous, 30% binary, 10% categorical) and 10% injected missingness to simulate realistic scenarios.

The SIGMA framework is implemented through a three-stage pipeline: (1) learning the structural posterior via Flow-SEM and sampling 1,000 DAGs; (2) identifying high-confidence mediation pathways based on Path Stability Score (threshold τ = 0 . 15 ); (3) estimating natural direct and indirect effects for identified pathways, with estimates aggregated across multiple DAGs via Bayesian model averaging to propagate structural uncertainty. Effect estimation employs a 5-fold cross-fitting strategy, utilizing EIF for paths of length 3 and plug-in estimators for longer paths.

5.1 Causal Discovery Evaluation

Datasets. We generate 10 synthetic datasets with varying characteristics, including different graph sizes from low to high dimensional p ∈ { 20 , 50 , 100 , 200 } , nonlinearity level ( ρ nonlin ∈ { 0 , 0 . 5 } ), graph structures (parallel κ ∈ { 2 , 3 , 6 } and chain ℓ ∈ { 1 , 2 } ), and variable types variables (continuous, binary, categorical). Dataset-specific configurations are provided in Appendix E.3.

Figure 2 shows that SIGMA demonstrates consistently strong performance across datasets. On datasets (HighDim-D/P), DECI has a slight advantage in adjacency F1 (0.3814/0.4869 vs. 0.3741/0.4754), potentially due to differences between SIGMA's Flow-SEM continuous optimization framework in capturing weak connections under high-dimensional sparse conditions and DECI's global variational inference approach. In contrast, SIGMA shows comprehensive leadership in orientation determination (average improvement of 42.1%), a capability crucial for accurate identification of mediation pathways. Experimental results confirm that SIGMA can recover directed structures under diverse conditions, providing a robust foundation for downstream causal effect estimation.

(a)

(b)

(c)

LowDim-D

LowDim-P

LowDim-D

LowDim-P

LowDim-D

LowDim-P

0.45

0.56

0.13

0.28

0.57

0.67

MidDim-S

LowDim-N

MidDim-S

LowDim-N

MidDim-S

LowDim-N

0.49

0.49

0.32

0.26

0.68

0.64

MidDim-D0.50

0.77LowDim-L

0.40

0.59

0.19

0.39

0.64

0.73

HigDim-S

0.37

0.48

HigDim-C

HigDim-S

0.18

0.32

HigDim-C

HigDim-S

0.66

0.69

HigDim-C

HigDim-D

HigDim-P

HigDim-D

HigDim-P

-WIab!H

HigDim-P

PC

Notears

ICALiNGAM

Grandag

DECI

SIGMA

5.2 End-to-end Causal Mediation Analysis

We evaluate SIGMA's effect estimation performance on synthetic data by comparing estimates against ground truth values generated via structural equation models (see Appendix F.2). For comparison, we

use the Lasso, random forest (RF), gradient boosted machine (GBM) and DeepMed [65]. Two highconfidence mediation pathways were selected from the synthetic dataset LowDim-L for evaluation. As shown in Table 1, SIGMA consistently produces low bias and narrow confidence intervals across both natural direct and indirect effects. These results indicate that by sampling multiple DAGs from the structural posterior and applying BMA, SIGMA effectively incorporates both within-graph estimation uncertainty and across-graph structural variation. This leads to robust and stable inference of mediation effects under structural uncertainty.

Table 2 reports the performance metrics of SIGMA across different dimensionalities. As feature dimensionality increases from 500 to 2000, all three core evaluation metrics exhibit a declining trend: Adjacency F1 decreases by 22.5% (from 0.603 to 0.467), Orientation F1 decreases by 25.9% (from 0.541 to 0.401), and Path F1 decreases by 26.9% (from 0.587 to 0.429). Despite this performance degradation, SIGMA maintains an edge recovery F1 of 0.467 and a pathway identification F1 of 0.429 in the 2000-dimensional configuration, with all standard errors controlled within 0.083. These results confirm the scalability of the SIGMA framework for high-dimensional mediation analysis.

5.3 Real Data HPP Validation

Validation on the HPP dataset shows that the SIGMA framework successfully identified multiple physiologically meaningful mediation pathways (Figure 3). Specifically, the pathway (sleep HRV index → pulse wave velocity → carotid IMT) verified a known mechanism whereby sleep quality affects cardiovascular health via vascular function. Previous studies have reported associations between reduced sleep quality and autonomic dysfunction, indicated by decreased heart rate variability (HRV), which is linked to increased arterial stiffness (PWV) and subsequently greater carotid IMT [11, 32, 51]. Additionally, SIGMA also identified a pathway where sleep snoring severity affects carotid IMT via body fat percentage and blood pressure regulation. This suggest an indirect effect of sleep quality on vascular health mediated through metabolism and blood pressure. Existing clinical studies have linked severe snoring and obstructive sleep apnea (OSA) to obesity via endocrine disturbances [24]. Obesity, particularly increased visceral fat, has been linked to elevated blood pressure [73], and sustained high blood pressure is associated with increased carotid IMT [37]. Moreover, as illustrated in Figure 3, the analysis reveals associations among nocturnal hypoxic burden, OSA, body fat composition, carotid IMT, and gut microbiota abundance. This newly identified mediation pathway suggests connections among sleep-disordered breathing, metabolism, cardiovascular health, and gut microbiota ecology. This indicates that abnormal body fat distribution induced by sleep disturbances may further affect cardiovascular health through elevated blood pressure and autonomic dysfunction. Classical medical regression mediation analysis further validated these pathways, with an overall TE error of 0.2734 and consistent NIE and NDE estimation directions. Numerical results and visualization details are provided in the Appendix F.4.

0.2

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abundance

6 Related Work

CMAhas evolved from estimating total treatment effects to decomposing pathways into direct and indirect components [27, 45, 49]. Semiparametric methods leveraging EIFs have been widely adopted for this purpose, offering multiple robustness and asymptotic efficiency [59, 71]. Recent work such as DeepMed [65] advances this line by using deep neural networks to estimate high-dimensional nuisance functions without sparsity constraints, achieving semiparametric efficiency under weaker smoothness assumptions. However, these methods generally assume a known mediation structure. Parallelly, causal discovery frameworks like NOTEARS [72], GES [56], and DAG-GNN [69] aim to recover causal graphs from observational data but do not target mediation-specific pathways. DECI [22] proposes a variational approach that jointly infers causal structure and structural equations, supporting downstream inference via interventional queries. While powerful, DECI and similar methods [15, 60] focus on recovering global structure and interventional distributions rather than quantifying pathway-specific mediation effects. Other works in high-dimensional mediation [7, 18] have explored multi-mediator identification but typically rely on predefined structures or linear assumptions. This disconnect between causal discovery and effect estimation limits their applicability in complex, uncertain systems. Please refer Appendix C for detailed literature review.

7 Conclusion, Limitations and Future

The SIGMA framework we proposed provides an end-to-end solution for causal mediation analysis under structural uncertainty by unifying structure discovery, pathway identification, and effect estimation. The core innovation of this framework lies in: quantifying causal graph uncertainty through probabilistic structural posterior learning, automatically identifying high-confidence mediation pathways via the Path Stability Score, and systematically propagating structural uncertainty to final effect estimates through Bayesian model averaging. In validation experiments on HPP data, SIGMA confirmed known sleep-cardiovascular mediation relationships and discovered previously unspecified cross-system mediation pathways, revealing how sleep quality affects cardiovascular health through inflammatory and metabolic pathways, providing new targets for clinical intervention. However, as an observational research method, SIGMA remains limited by unmeasured confounders and selection bias. To address these challenges, we plan to conduct clinical intervention experiments, validating the inflammatory and metabolic mediation mechanisms discovered by SIGMA through sleep quality interventions (e.g., CPAP treatment, sleep hygiene improvement), providing stronger causal evidence, and developing sensitivity analysis tools to systematically assess the robustness of inference results, thereby facilitating translation from computational discoveries to clinical applications.

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NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The abstract and introduction of this paper precisely articulate the core contributions and intended scope of our proposed SIGMA framework. We clearly define SIGMA as an innovative end-to-end mediation pathway inference method that unifies structure discovery, pathway identification, and effect estimation, with a strong emphasis on its unique capability to systematically handle structural uncertainty. These claims are fully consistent with the advanced methodology and comprehensive experimental results detailed in the subsequent sections, accurately reflecting the value of our work.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We discuss the potential limitations of our work in Appendix A provide an in-depth exploration of the specific considerations and strategies for addressing core causal assumptions (e.g., Causal Markov, Faithfulness, and Causal Sufficiency) within the SIGMA framework in Appendix A.2 "Robustness Considerations and Sensitivity Analysis of SIGMA." This demonstrates our careful consideration of the applicability boundaries of our method and directions for future improvements.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: This paper provides a solid theoretical foundation for all theoretical results. We systematically elaborate on the definitions, notation, and key identification assumptions (including consistency, ignorability, positivity) for causal mediation analysis.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We are committed to ensuring high reproducibility of our experimental results. We believe this sufficient and transparent information provides a solid foundation for other researchers to reproduce our main experimental results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: We have thoroughly described the generation methods and parameters for the synthetic data in Appendix, ensuring the reproducibility of these experiments. For the HPP dataset, we have provided a link to its official knowledge base (https://knowledgebase.pheno.ai/) for reference.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We have provided sufficient experimental setting details to ensure readers can understand and evaluate our results.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We have given due attention to the statistical significance of our experimental results in the paper. Specifically, we clearly reports the bias, empirical standard error (SE), root mean squared error (RMSE), and the coverage probability (CP) of their corresponding 95% confidence intervals for the estimated Natural Direct Effect (NDE) and Natural Indirect Effect (NIE). These metrics provide robust statistical evidence for assessing the accuracy and stability of our method's estimates.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We discuss in Appendix.

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: We have strictly adhered to the NeurIPS Code of Ethics in conducting this research. This study utilizes data from the Human Phenotype Project (HPP), which received formal approval from the Institutional Review Board (IRB) of the Weizmann Institute of Science.Furthermore, all participants provided written informed consent. This ensures that our research aligns with recognized ethical standards in all respects.

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We provide a preliminary discussion of the potential impacts of our work.As a foundational research framework designed to uncover causal mechanisms in complex systems, SIGMA holds broad positive application potential in fields such as biomedicine, public health, and social sciences, for instance, by aiding in the understanding of disease mechanisms or evaluating the effects of policy interventions. While our primary focus is on methodological innovation, we also recognize that any powerful analytical tool must be used responsibly.

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The SIGMA framework proposed in this paper is a causal inference methodology and, as such, does not constitute a model type with a high risk of misuse (e.g., large-scale language models or image generation models). The HPP dataset we utilized is biomedical research data that has undergone rigorous ethical review, not a dataset scraped from the internet or known to have high inherent risks. Therefore, specific safeguards for such models or data are not applicable to this research. account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We strictly adhere to academic standards by appropriately crediting and citing the creators or original owners of all existing assets (including algorithms, datasets, and software tools) used in this paper, as detailed in our References section. For instance, for the HPP dataset, we have clearly indicated its origin and cited relevant primary research literature.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: This paper itself (including the main text and appendices) serves as comprehensive and detailed documentation for our innovative SIGMA framework. We have systematically elucidated all components of the SIGMA framework, its solid theoretical underpinnings, a clear algorithmic workflow, and thorough experimental validation within the manuscript. This content provides a complete guide for understanding and applying the SIGMA framework.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: This research did not involve any crowdsourcing experiments. We utilized the existing HPP dataset, which includes human subjects. As detailed in Appendix, the HPP study itself adhered to strict ethical procedures, including written informed consent from all participants. As a secondary analysis of the HPP data, this paper did not directly recruit or interact with subjects, and therefore, providing additional instructions or compensation to participants was not part of our research protocol. Relevant information pertaining to the original HPP study is detailed in its respective protocols.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [Yes]

Justification: We place a high priority on research ethics. The Human Phenotype Project (HPP) study protocol, from which our data is derived, received formal approval from the Institutional Review Board (IRB) of the Weizmann Institute of Science. Furthermore, all participants in the HPP study provided written informed consent, implying that potential risks of the study were fully disclosed to the subjects, and their consent was obtained. This paper strictly conducts data analysis within this ethical framework.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: Large Language Models (LLMs) were not utilized as an important, original, or non-standard component in the core methodology of this research. Our research focuses on the development and validation of the SIGMA framework, which is based on causal

inference techniques such as structural equation models, path stability scoring, and Bayesian model averaging. LLMs were not employed in the core algorithm design or experimental analysis of this study.

A Detailed Discussion of Causal Assumptions

A.1 Core Assumptions for Causal Structure Discovery and Identification

The validity of causal structures inferred by Flow-SEM-and, by extension, the SIGMA framework-relies on a set of core assumptions that connect observed statistical dependencies to underlying causal mechanisms. In particular, the following three foundational assumptions enable reliable identification and interpretation of causal graphs from observational data.

A.1.1 The Causal Markov Assumption

Formal Definition. A probability distribution P ( V ) is said to satisfy the Causal Markov Condition with respect to a DAG G = ( V, E ) if each variable X i ∈ V is conditionally independent of its non-descendants, given its parents in the graph. Formally, for all X i ∈ V :

where Pa G ( X i ) denotes the set of parent nodes of X i , and NonDesc G ( X i ) denotes its nondescendants in G . For DAGs, this local condition is equivalent to the global Markov condition defined via d-separation : for disjoint subsets X,Y,Z ⊆ V , if Z d-separates X and Y in G , then X ⊥ ⊥ Y | Z in P ( V ) .

Role in SIGMA. The Causal Markov Assumption justifies the structural factorization used by Flow-SEM in SIGMA. Specifically, it enables the modeling of the joint distribution P ( V | W θ , Φ ) as the product of conditionals:

which underpins the Structural Equation Model (SEM) formulation:

where f i and σ i are flexible functions (e.g., neural networks), and U i ∼ N (0 , 1) represents exogenous noise. This factorization supports the likelihood-based training objective used in Flow-SEM (see Section 4.2.1). In particular, Equation 3 optimizes the negative log-likelihood while enforcing acyclicity and sparsity constraints. Without the Causal Markov assumption, the inferred structure may not reflect the true causal dependencies, even when the model achieves high likelihood under the observed data distribution.

A.1.2 The Faithfulness Assumption

Formal Definition. A distribution P ( V ) is faithful to a DAG G = ( V, E ) if every conditional independence observed in P ( V ) corresponds to a d-separation in G . Formally, for any disjoint subsets X,Y,Z ⊆ V :

The forward implication ( ⇒ ) corresponds to the Causal Markov Assumption, while the reverse ( ⇐ ) constitutes the Faithfulness Assumption.

Role in SIGMA. Faithfulness is critical to ensuring that Flow-SEM accurately infers the presence or absence of causal edges. Violations occur when distinct causal pathways counteract each other such that their net effect cancels out. For instance, in competing mediation paths:

which can induce conditional independencies not reflected by any d-separation in the true DAG. While such exact cancellations are measure-zero events in parameter space, they can still arise in practical scenarios.

Violations of faithfulness in SIGMA result in biased structure learning. Specifically, the learned posterior P ( G s | D ) may diverge from the true posterior over the causal graph G ∗ :

̸

leading Flow-SEM to favor incorrect structural hypotheses. This has downstream effects on mediation pathway identification, particularly through the computation of the Path Stability Score (PSS), defined as:

where π denotes a candidate mediation path, and the indicator function returns 1 if all directed edges in π exist in the sampled DAG G s . As a result, true causal pathways may be assigned artificially low PSS values due to systematically missing edges, reducing their chances of being selected for downstream effect estimation.

A.1.3 The Causal Sufficiency Assumption

Formal Definition. A variable set V obs is causally sufficient if for every pair X,Y ∈ V obs , all common causes are also in V obs. Equivalently, no unmeasured variable U / ∈ V obs is a direct cause of two or more variables in V obs .

Role in SIGMA. Causal sufficiency plays a pivotal role across all components of SIGMA. In the structure discovery phase, unmeasured confounders can distort edge estimation, leading to divergence between the learned and true graph posteriors:

̸

where G ∗ denotes the true underlying causal graph.

In the effect estimation phase (Section 4.4), causal sufficiency affects the correctness of nuisance function estimation. These functions, which are essential for computing NDE and NIE, depend on the appropriate conditioning sets defined by the graph:

When G s is misspecified due to unmeasured confounding, the adjustment sets are invalid, thereby violating key identification assumptions such as:

SIGMA assumes that the full variable set V = { A,Y } ∪ M ∪ C is causally sufficient, or that any substantial unobserved confounding is accounted for within C . When this assumption is questionable, violations can be partially mitigated through sensitivity analysis.

A.1.4 SIGMA Framework to Challenges in Core Causal Assumptions

The efficacy of inferring causal structures G = ( V, E ) and quantifying mediational pathway effects from observational data D is intrinsically linked to a set of foundational causal assumptions. The SIGMA framework incorporates specific mechanisms in its design to consider and respond to the challenges these assumptions may face in complex real-world systems.

Considerations regarding the Causal Sufficiency Assumption

The Causal Sufficiency assumption, requiring that all common causes U affecting any two variables V i , V j ∈ V within the system are included in the observed set V (i.e., U ⊆ V ), is an ideal yet often challenging prerequisite for unbiased causal effect estimation. When this condition is potentially violated, implying the existence of unmeasured confounders U / ∈ V , SIGMA offers a more robust analytical pathway through the following design considerations:

Structured Integration of High-Dimensional Covariates: SIGMA mandates the explicit inclusion of a set of covariates C within the observed variable set V = { A,Y } ∪ M ∪ C (see Section 3 of the

main text). The Flow-SEM module (Section 4.2.1 of the main text) incorporates these covariates C into the structural learning process, treating them equivalently to other variables ( A,M k ∈ M,Y ). This allows for the data-driven discovery of relationships between C and A,M,Y . By employing flexible functions f i ( Pa G ( V i ); ϕ i ) (Equation 1 of the main text), SIGMA can capture complex, non-linear confounding patterns mediated by the observed covariates C , thereby adjusting for such confounding at a structural level, which surpasses traditional methods that often treat C merely as linear control variables.

Exploration of Structural Posterior Distribution: Rather than relying on a single, potentially misspecified, predefined causal graph, SIGMA leverages Flow-SEM to learn a parameterized posterior distribution P ( G | D, W θ , Φ ) over causal structures. From this distribution, an ensemble of DAGs G = { G s } N DAG s =1 is sampled (Section 4.2.3 of the main text). This ensemble G represents a multiplicity of causal structure hypotheses compatible with the observed data D and the given variable set V . This strategy acknowledges that, even conditional on V , the true causal structure may be uncertain. Different G s ∈ G may account for complex dependencies arising from C in various ways, thus providing a more comprehensive examination of confounding effects mediated by observed variables.

Structure-Guided Bayesian Model Averaging: For each candidate pathway π identified via the Path Stability Score (PSS) (Section 4.3 of the main text), SIGMA estimates pathway-specific effects ˆ θ s ( π ) under each supporting DAG G s ∈ G π ⊆ G . The nuisance functions ˆ η s are parameterized based on the parent sets Pa G s ( · ) (including relevant covariates from C ) defined by that specific G s (Sections 4.4.1, 4.4.2 of the main text). The final effect estimate ˆ θ BMA ( π ) and its variance ˆ V ( ˆ θ BMA ( π )) are obtained by BMA over G π (Section 4.4.3, Equation 7 of the main text). This aggregation ensures that the final inference incorporates not only the statistical uncertainty ˆ V ( ˆ θ s ( π )) within each specific structure but also the structural uncertainty, Var G s ∈G π [ ˆ θ s ( π )] , arising from the heterogeneity of causal hypotheses consistent with the observed data V .

Although SIGMA cannot intrinsically detect or fully neutralize biases from truly unmeasured confounders (i.e., those not included in V ), its design aims to maximize the utility of all available information within V (especially covariates C ). Through structured modeling of observed covariates, explicit modeling of structural uncertainty, and aggregation over multiple structural hypotheses, SIGMA provides a more systematic and data-driven framework than analyses relying on a single, fixed causal model to mitigate biases stemming from confounding by observed variables. For potential unmeasured confounding beyond the scope of V , sensitivity analysis, as recommended in Appendix A.1.3 of the main paper, remains an important complementary tool for assessing the robustness of conclusions.

Considerations regarding the Causal Faithfulness Assumption

The Causal Faithfulness assumption posits that all conditional independence relations present in the probability distribution P ( V ) are entailed by d-separations in the true causal graph G ∗ , and vice-versa (Sections 3, A.1.2 of the main text). This implies the absence of statistical independencies arising from exact cancellations of path effects (e.g., ∑ π ∈ Π A ⇝ Y effect ( π ) ≈ 0 while Π A ⇝ Y = ∅ ), where underlying causal pathways exist. SIGMA's mechanisms offer the following considerations when facing challenges of approximate or exact unfaithfulness:

̸

Continuous Edge Weight Learning and Probabilistic Representation: Flow-SEM (Section 4.2.3 of the main text) learns a continuous adjacency weight matrix W θ , where entries | [ W θ ] ji | reflect the strength of a potential edge V j → V i . These weights are transformed into edge inclusion probabilities p ji = P ( e ji ∈ E | D ) via σ ( | [ W θ ] ji | ) (Section 4.2.3 of the main text). In scenarios of approximate unfaithfulness, where the net effect of a path is close to zero, diminishing the signal for its constituent edges, the corresponding p ji values may be low but not necessarily zero (unless strong sparsity priors are imposed). This allows edges with weak signals a non-zero probability of being included in sampled DAGs, thus mitigating premature exclusion based on hard independence thresholds.

Statistical Filtering via Path Stability Score (PSS): The PSS ( π ) (Section 4.3, Equation 5 of the main text) is defined as the frequency of path π 's appearance across N valid sampled DAGs, G s , serving as a Monte Carlo estimate, E G ∼ P ( G | D ) [ I ( π ⊆ G )] , of its marginal inclusion probability under the learned posterior P ( G | D ) . If a true path's signal is attenuated due to approximate unfaithfulness affecting one of its constituent edges e ji (resulting in a low p ji ), the joint probability of observing π (and thus its PSS ( π ) ) will be reduced. By thresholding PSS ( π ) with τ , SIGMA selects paths that

exhibit sufficient statistical consistency across the ensemble, effectively focusing on pathways whose structural evidence remains discernible despite data noise and potential complexities in parameter configurations.

Diversity of Structural Ensemble and BMA Weighting: SIGMA generates a diverse ensemble of DAGs G . Even if certain parameter configurations lead to the apparent weakness or absence of a specific path in some G s ∈ G due to approximate unfaithfulness, such a path might still manifest in other G s corresponding to different plausible parameterizations, provided some evidential support exists in D . While the BMA procedure in this work employs uniform weights for selected paths, the PSS acts as a preceding filter. Paths consistently undermined by strong approximate unfaithfulness, resulting in PSS values below τ , are excluded from the final effect aggregation. This prioritizes mediational mechanisms with more robust and identifiable signals in the observed data.

Consequently, while SIGMA cannot resolve fundamental unidentifiability stemming from perfect parameter cancellations, its probabilistic and ensemble-based framework allows for a more nuanced, data-driven assessment of pathways whose signals might be weakened by approximate unfaithfulness. It quantifies the evidential strength for each path via PSS, thereby focusing inferential resources on mediational mechanisms that are statistically stable and discernible under the learned posterior distribution of structures.

Considerations regarding the Causal Markov Condition

The Causal Markov Condition, a cornerstone of graphical causal models, states that any variable V i is conditionally independent of its non-descendants (excluding its parents) given its direct causes Pa G ( V i ) in a DAG G (Section A.1.3 of the main text). SIGMA's design inherently relies on and leverages this assumption:

DAG-based Parametrization: The Flow-SEM module (Section 4.2.1 of the main text), central to SIGMA's structure discovery, models inter-variable relationships using a Directed Acyclic Graph (DAG). The structural equations V i = f i ( Pa G ( V i ); ϕ i ) + σ i ( Pa G ( V i ); ϕ i ) · U i (Equation 1 of the main text) explicitly encode that the distribution of V i is determined solely by its parents Pa G ( V i ) and an exogenous noise term U i .

Factorization of Likelihood Function: The log-likelihood function log P ( V ( j ) | W θ , Φ ) in FlowSEM (Equation 2 of the main text) is expressed as a sum over terms corresponding to each variable V i conditioned on its parents Pa G ( V ( j ) i ) , i.e., ∑ p i =1 ( log p U ( u ( j ) i ) -log σ i ( Pa G ( V ( j ) i ); ϕ i ) ) . This factorization is a direct mathematical consequence of the Causal Markov Condition, rendering the learning of high-dimensional joint distributions computationally tractable.

Flexible Functional Forms and Structural Exploration: By employing neural networks for f i and σ i , SIGMA accommodates complex non-linear dependencies. The learning of W θ and subsequent sampling of the DAG ensemble G allow SIGMA to explore a rich space of DAG structures that are compatible with the data and inherently satisfy the Markov condition. This combination enables SIGMA to seek the best description of observed data dependencies within the class of Markovian models representable by DAGs.

A.2 Robustness Considerations and Sensitivity Analysis of SIGMA

The theoretical guarantees for SIGMA framework validity rely on a series of core causal assumptions, including the Causal Markov condition, Faithfulness, and Causal Sufficiency defined in Appendices A.1, as well as effect identification conditions such as Ignorability and Positivity specified in Section 3 of the main text. In practical application contexts, these assumptions may be violated to varying degrees, threatening the validity and reliability of the estimated mediational pathway effects. Therefore, systematic analysis of the robustness and sensitivity of results obtained under the SIGMA framework becomes particularly necessary. Next, we discuss potential methodological frameworks for sensitivity analysis and mathematical prospects for future research.

A.2.1 Sensitivity Analysis under Deviations from Causal Sufficiency

Consider an unobserved potential confounding variable U that simultaneously affects exposure A , the set of mediator variables M , and the outcome variable Y . In the presence of such unobserved confounding, we define a confounding sensitivity analysis framework modulated by U as follows:

Assume the confounding variable U is simultaneously associated with exposure and mediators, as well as outcomes and mediators, with their association strengths described by the following two sensitivity parameters Γ AU , Γ UY :

Here Γ AU , Γ UY ≥ 1 , with larger values indicating stronger influence of the unobserved confounding variable. To quantify the robustness of observed effects to the strength of the above unobserved confounding, the E-value [62] can be used, defined as:

where CI Γ AU , Γ UY ( ˆ θ BMA ( π )) represents the confidence interval for the effect estimate adjusted for U given the sensitivity parameters. A larger E-value indicates that the conclusion is more robust to unobserved confounding; conversely, a smaller E-value indicates weaker robustness. Future research could consider further integrating the above sensitivity analysis automatically into the SIGMA process, using simulation studies to quantify its sensitivity performance under various data-generating mechanisms.

A.2.2 Robustness Analysis under Deviations from Causal Faithfulness

Suppose the data-generating mechanism has approximate effect cancellations, where the true effect of some path π in the path set Π A → Y is θ ( π ) ≈ 0 , despite this path existing in the true causal structure (i.e., approximate unfaithfulness). This scenario leads to the estimated Path Stability Score (PSS, Section 4.3) tending toward lower values. Sensitivity analysis considers a path stability threshold τ ∈ [0 , 1] , defining a stable path set as:

By studying the impact of changes in τ on the final Bayesian Model Averaging (BMA) estimate ˆ θ BMA ( π | τ ) , we can examine the sensitivity of ˆ θ BMA ( π | τ ) to different τ :

In applications, if low sensitivity is found (the above derivative is small), it indicates that the SIGMA estimation conclusion is relatively robust; conversely, if high sensitivity is found, it may suggest that approximate unfaithfulness significantly affects the analysis, requiring cautious interpretation. Future research could deeply explore the properties of this derivative and its theoretical and empirical performance under different structural assumptions.

A.2.3 Robustness Analysis of Model Specification Errors in Effect Estimation

When estimating path-specific effects ˆ θ BMA ( π ) , SIGMA involves multiple nuisance functions (e.g., propensity scores and conditional expectation functions). Setting the bias of these nuisance function estimates as δ , the following form of bias sensitivity analysis can be considered:

Define the sensitivity function of estimator bias to nuisance function estimation bias:

Lower sensitivity values imply higher robustness of the estimator to model specification errors, and vice versa. The Cross-Fitting strategy adopted in SIGMA provides a certain degree of robustness guarantee, but future research should systematically compare the sensitivity function characteristics of this estimation strategy with other semiparametric doubly robust estimators.

A.2.4 Interplay

The three assumptions-Causal Markov, Faithfulness, and Causal Sufficiency-jointly define the identification conditions for reliable causal discovery in SIGMA. The Causal Markov condition links the graphical structure to conditional independence relations. The Faithfulness assumption ensures that all observed independencies reflect actual graphical separations rather than coincidental parameter cancellations. Causal Sufficiency guarantees that all relevant common causes are observed, preventing bias from unmeasured confounding. Together, these assumptions enable SIGMA to learn meaningful causal structures from observational data and propagate this structure into consistent mediation effect estimation.

A.3 Identification Assumptions for NDE and NIE

TThe nonparametric identification of Natural Direct Effects (NDE) and Natural Indirect Effects (NIE) requires a set of conditional independence assumptions that account for confounding. These effects are defined in the potential outcomes framework (see Section 3) as: NDE ( a, a ∗ ) = E [ Y ( a, M ( a ∗ )) -Y ( a ∗ , M ( a ∗ ))] and NIE ( a, a ∗ ) = E [ Y ( a, M ( a )) -Y ( a, M ( a ∗ ))] , These assumptions ensure that observed statistical associations admit a valid causal interpretation. In the context of SIGMA, this is particularly important because the underlying causal graph G ∗ is unknown and approximated via a sampled ensemble { G s } .

A.3.1 Ignorability Assumptions for NDE and NIE Identification

A common set of sufficient conditions for the identification of NDE and NIE, often referred to as "sequential ignorability" or "no unmeasured confounding" assumptions, can be stated conditional on a set of pre-treatment covariates C [45, 8, 49]. For treatment levels a and a ∗ :

Conditional independence of treatment and potential outcomes/mediators (Controlling for A-M and A-Y confounding): There is no unmeasured confounding of the effect of A on M , and no unmeasured confounding of the effect of A on Y that is not mediated by M . Formally:

Together, these imply that C blocks all back-door paths from A to M , and from A to Y that do not go through M . A common shorthand formulation is:

Conditional independence of mediator and potential outcomes, given treatment and covariates (Controlling for M-Y confounding): This assumption requires that, conditional on treatment A and covariates C , there is no unmeasured confounding between the mediator M and the outcome Y :

Under this assumption, among units with the same treatment and covariates, the mediator behaves as if it were randomly assigned with respect to the counterfactual outcome Y ( a, m ′ ) .

Conditional independence of treatment and potential outcomes, given mediator and covariates (No treatment-induced M-Y confounding that also depends on A directly): This assumption rules out any variable that is affected by treatment and also confounds the relationship between M and Y . In some formulations, this condition is written as:

However, this expression involves cross-world counterfactuals , which are often avoided in practice. Instead, most identification strategies rely on single-world assumptions derived from the causal graph.

For example, the mediation formula [45] provides a means to identify NIE using the following expression:

This formula is valid under Assumptions 1 and 2, as well as the Causal Markov and Faithfulness assumptions for the underlying DAG encoding the relationships among A,M,Y,C . Within the SIGMA framework, pathway-specific effect estimation relies on Efficient Influence Functions (EIFs), which are computed relative to each sampled graph G s . For example, to estimate the effect along a mediation pathway A → M j → Y , SIGMA computes the following nuisance quantities:

These estimates rely on graph-structured adjustment sets and implicitly assume that the corresponding back-door paths are blocked. As such, the validity of EIF-based identification hinges on the correctness of the local parent sets derived from G s . In cases where no variable is jointly affected by A and also acts as a confounder of the M → Y link, the stronger cross-world independence condition Y ( a ∗ , m ) ⊥ ⊥ M ( a ) | C can be avoided [49].

Role of Graphical Structure G s in Satisfying Ignorability within SIGMA. In SIGMA, the Flow-SEM module generates an ensemble of candidate DAGs { G s } . For each graph G s = ( V, E s ) , the estimation of nuisance functions ˆ η s (see Section 4.4.1), such as the conditional expectation E [ Y | A,M,Z Y ] and the conditional probability P ( M | A,Z M ) , relies on structure-specific conditioning sets. These sets are constructed from the local parent sets Pa G s ( · ) defined by each DAG, together with the baseline covariates C . Specifically, we define:

This formulation reflects the assumption that causal relationships are encoded in G s , and that proper conditioning on parent sets and covariates is sufficient to block non-causal pathways. The theoretical justification rests on Pearl's d-separation criterion and the back-door adjustment principle [46].

Under ignorability Condition 1, which requires A ⊥ ⊥ M ( a ′ ) | C and A ⊥ ⊥ Y ( a ′′ , m ) | C , a set of variables-denoted C ′ -must block all back-door paths from A to M , and from A to Y that are not mediated through M . These paths must begin with an arrow into A , and blocking them ensures that the necessary conditional independencies hold in G s . For Condition 2, which assumes M ⊥ ⊥ Y ( a, m ′ ) | A = a, C , the relevant conditioning set must block all back-door paths between M and Y , conditional on A = a . This is achieved by the set C ∪ Pa G s ( Y ) \ { A,M } ∪ Pa G s ( M ) \ { A } , where conditioning on A = a accounts for any paths passing through A as either a collider or a non-collider.

The central premise is that each sampled DAG G s is a candidate for the true underlying causal graph G ∗ , and that the observed variable set V obs = { A,Y } ∪ M ∪ C satisfies causal sufficiency. When these conditions hold, and if G s accurately reflects the structural relationships encoded in G ∗ , then using the parent sets Pa G s ( · ) in conjunction with C provides a principled approach for selecting conditioning variables based on the d-separation criterion. For instance, if Pa G s ( M ) includes all common causes of M and Y that are not affected by A and are not in C , then conditioning on A,C, Pa G s ( M ) \ { A } is sufficient to satisfy the M-Y ignorability condition:

However, since each G s is only a hypothesis, ignorability is not guaranteed unless G s = G ∗ and the covariate set C is truly sufficient. If either the graph is misspecified or unmeasured confounders are omitted from both C and the parent sets, the resulting conditioning set C ∪ Pa G s ( · ) may be inadequate. This misalignment can lead to biased nuisance function estimates and consequently biased causal effect estimates ˆ θ s ( π ) . While the BMA procedure in SIGMA propagates structural uncertainty by averaging across sampled DAGs, it does not inherently resolve violations of ignorability that are present across all candidate structures. Therefore, rigorous selection of covariates C remains essential for ensuring the validity of effect estimation under structural uncertainty.

A.3.2 Consistency Assumption

The consistency assumption establishes the fundamental link between potential outcomes (defined under hypothetical interventions) and observed outcomes in the data. It posits that for any individual,

the observed outcome corresponds precisely to the potential outcome associated with the treatment and mediator values that the individual actually received.

Formally, for all individuals i , the assumption requires:

which means that if an individual received treatment a , the observed mediator equals the potential mediator under a ; and if the individual received treatment a and mediator value m , the observed outcome equals the potential outcome under those values.

These conditions can be restated as follows. For all individuals i , and any treatment level a ∈ A , if A i = a , then M i = M i ( a ) . This expresses that the observed mediator is what would have been realized under the assigned treatment. Similarly, for all a ∈ A and all m ∈ M , if A i = a and M i = m , then Y i = Y i ( a, m ) . This implies that the observed outcome reflects the potential outcome under the observed treatment and mediator values. These requirements must hold for all relevant treatment levels (e.g., a , a ∗ in the definition of NDE and NIE) and all values of m within the support of M .

Consistency is essential for expressing counterfactual estimands such as E [ Y ( a, M ( a ∗ ))] in terms of observable quantities. Under this assumption, conditional expectations from the observed distribution P ( Y, A, M, C ) can be equated to expectations over potential outcomes. For example, the observed expectation E [ Y | A = a, M = m,C = c ] corresponds to E [ Y ( a, m ) | A = a, M = m,C = c ] , assuming that ignorability holds for the conditioning set.

In the SIGMA framework, consistency enables the interpretation of structure-specific nuisance functions as counterfactual components. These include, for each sampled DAG G s :

which are estimated from observational data and treated as valid components of EIF-based identification strategies (see Section 4.4.1). This directly affects single-DAG effect estimates ˆ θ s ( π ) (Section 4.4.2) and, through Bayesian model averaging, the final estimates ˆ θ BMA ( π ) .

Violations of consistency occur when the treatment A = a or mediator M = m is ambiguously defined in the data. For example, if multiple unobserved variants of A = a exist (e.g., 'exercise intervention' encompassing both 'light jogging' and 'intense weightlifting'), and these variants have distinct effects on downstream variables, then consistency does not hold, even though the data records only a single treatment label. The same logic applies if M is coarsely defined.

SIGMA implicitly assumes that treatment A and mediator M are sufficiently well-defined and measured with adequate granularity such that their causal interpretation is meaningful. Ensuring consistency is not typically addressed at the algorithmic level, but rather requires careful study design or strong domain knowledge to define, operationalize, and measure treatments and mediators in a causally coherent way.

A.3.3 Positivity Assumption

The positivity assumption, sometimes referred to as overlap or the experimental treatment assignment (ETA) assumption, is essential for the nonparametric or semiparametric identification of causal effects, including NDE and NIE. It states that, for any combination of observed pre-treatment covariates C = c , each treatment level A = a and each mediator value M = m must occur with non-zero probability. This ensures adequate empirical support across all necessary strata, which is required to estimate causal contrasts without extrapolation.

Let A denote the set of treatment values, M the support of the mediator(s) M , and C the support of covariates C . The positivity condition for NDE and NIE identification can be stated as:

where ϵ A > 0 and ϵ M > 0 are small constants ensuring strict positivity. For continuous mediators, the density p ( M = m | A = a, C = c ) must be bounded away from zero over the relevant support of M . In practical applications, especially when using estimators involving inverse probability weighting

(e.g., EIF-based estimators), stronger bounds such as ϵ A ≤ P ( A = a | C = c ) ≤ 1 -ϵ A are often imposed to ensure numerical stability.

In SIGMA, positivity is required for the estimation of nuisance functions ˆ η s (Section 4.4.1), such as the propensity score P ( A | Pa G s ( A ) , C ) and the conditional mediator distribution P ( M j | A, Pa G s ( M j ) \ { A } , C ) , for each sampled DAG G s . When Equation 29 or Equation 30 is violated in practice-i.e., some combinations of A,M,C occur with near-zero probability-the corresponding weights in EIFs or related estimators become excessively large. This can lead to inflated variance or unstable effect estimates ˆ θ s ( π ) .

For example, if p ( M = m | A = a, Pa G s ( M ) \ { A } , C = c ) is close to zero in some region of the covariate space, estimating expectations such as E [ Y | A = a, M = m,... ] requires extrapolation into areas with limited or no observed data. This weakens the reliability of the learned nuisance models and introduces instability in the resulting causal effect estimates.

These issues propagate to the final pathway-specific estimates through their effect on the single-DAG estimators ˆ θ s ( π ) , and subsequently affect the BMA-aggregated estimates ˆ θ BMA ( π ) . Although SIGMA employs flexible nuisance function models (e.g., neural networks), which may tolerate near-violations better than simpler parametric approaches, such models can still produce biased or high-variance estimates when positivity is violated. This underscores the importance of verifying covariate support overlap in practice and ensuring estimation occurs within sufficient empirical support.

Interaction with Other Assumptions. Positivity is intrinsically linked to other core assumptions:

Ignorability: Ignorability requires adjustment for a sufficient set of confounders C ′ (e.g., C ∪ Pa G s ( · ) ). Positivity ensures that such adjustment is empirically feasible by guaranteeing that all levels of A (and M ) are present across all strata of C ′ . Without Positivity, even if C ′ theoretically suffices for Ignorability, the lack of common support can make the adjustment ineffective or impossible (Appendix A.3.1).

Causal Sufficiency: If Causal Sufficiency is violated, the set C may be inadequate. Even if Positivity holds for the observed C , the inability to adjust for unobserved confounders means Ignorability may still be violated, leading to biased estimates irrespective of data overlap on C (Appendix A.1.3).

Practical Diagnostics for Positivity Violations. While SIGMA does not include automated procedures for testing positivity, several diagnostic practices are recommended to identify potential violations:

Examine Propensity Score Distributions: Plot histograms or density plots of estimated propensity scores P ( A = a | adjustment set ) for each treatment group. Lack of overlap or scores concentrated near 0 or 1 indicates potential Positivity issues.

Assess Covariate Balance: After stratification or weighting by propensity scores (if applicable to a diagnostic step), check for balance in covariate distributions across treatment groups using metrics like Standardized Mean Differences (SMDs). Large SMDs (e.g., > 0.1-0.25) can suggest poor overlap.

Inspect Weight Distributions: For estimators that explicitly use inverse probability weights (even internally, like some EIFs), examine the distribution of these weights. A few extremely large weights can dominate the estimate and inflate variance, signaling Positivity problems.

Check for Sparse Strata: For conditional mediator distributions P ( M | A, adjustment set ) , particularly with discrete mediators or high-dimensional adjustment sets, tabulate or examine the number of observations in critical ( A,M, adjustment set ) strata to identify sparse data regions.

Sensitivity Analysis: Evaluate how NDE/NIE estimates change if observations with extreme propensity scores are trimmed or if the analysis is restricted to regions of better covariate overlap (e.g., trimming based on propensity score tails).

Addressing severe positivity violations may require restricting the analysis to subpopulations with sufficient empirical support or applying estimation techniques designed for limited overlap (e.g., overlap weights or covariate balancing methods). These approaches can help reduce estimation

variance but may also introduce new assumptions or trade-offs, such as reduced generalizability or increased bias.

A.3.4 Single-DAG Effect Estimation via Influence Functions and Plugin Estimators

This appendix details the specific construction of Efficient Influence Functions (EIFs) for estimating the Natural Direct Effect (NDE) and Natural Indirect Effect (NIE) within the SIGMA framework when the exposure variable A is binary (taking values 0 or 1 ) and the path length is 3 (i.e., A → M → Y ). These EIFs depend on a series of nuisance function estimates obtained through K-Fold cross-fitting (see Section 4.4.1 in the main text for details). We define the following core nuisance functions, where the superscript ( s ) indicates that these functions are estimated for a specific DAG G s , but will be omitted in the following for brevity:

ˆ π ( C A ) = ˆ P ( A = 1 | C A ) : The propensity score for exposure A , i.e., the estimate of the conditional probability of A = 1 given covariates C A (parent nodes of A in G s ). For the mediator M (which can be continuous, binary, or categorical):

ˆ µ M ( a, C M ) = ˆ E [ M | A = a, C M ] : When M is continuous or binary, the estimate of its conditional expectation given A = a and covariates C M (parent nodes of M in G s , excluding A ).

ˆ f M ( m | a, C M ) = ˆ P ( M = m | A = a, C M ) : When M is discrete categorical, the estimate of its conditional probability mass function of taking value m given A = a and C M . For unified notation, we sometimes still use ˆ E [ g ( M ) | A = a, C M ] to denote the expectation of some function g ( M ) of M , which can be obtained by summing/integrating over ˆ f M .

ˆ µ Y ( a, m, C Y ) = ˆ E [ Y | A = a, M = m,C Y ] : The estimate of the conditional expectation of outcome Y given A = a , M = m , and covariates C Y (parent nodes of Y in G s , excluding A,M ).

Furthermore, we define the plug-in estimator for counterfactual expectations (obtained via Gcomputation using the aforementioned nuisance functions and averaged over the sample):

ˆ γ a ′ a ′′ = ̂ E [ Y ( a ′ , M ( a ′′ ))] = 1 n ∑ n i =1 ∫ ˆ µ Y ( a ′ , m, C Y,i ) d ˆ F M ( m | a ′′ , C M,i ) , where the integral sign represents summation over all possible values of the mediator M (discrete M ) or integration (continuous M ), and d ˆ F M ( m | a ′′ , C M,i ) represents the estimated conditional distribution based on ˆ µ M ( a ′′ , C M,i ) or ˆ f M ( m | a ′′ , C M,i ) .

NDE and NIE can be expressed as differences of these counterfactual expectations:

NDE = E [ Y (1 , M (0))] -E [ Y (0 , M (0))] = γ 10 -γ 00

NIE = E [ Y (1 , M (1))] -E [ Y (1 , M (0))] = γ 11 -γ 10

Therefore, the influence functions for NDE and NIE are linear combinations of the influence functions of their constituent counterfactual expectation terms. We present the influence function ψ γ a ′ a ′′ ( Z i ) for E [ Y ( a ′ , M ( a ′′ ))] (i.e., γ a ′ a ′′ ). For observed data Z i = ( A i , M i , Y i , C A,i , C M,i , C Y,i ) , the typical form of ψ γ a ′ a ′′ ( Z i ) is as follows:

Here, ˆ P ( A i = a ′ | C A,i ) is ˆ π ( C A,i ) when a ′ = 1 , and 1 -ˆ π ( C A,i ) when a ′ = 0 . When M is continuous, ˆ f M ( m | a, C M ) is the conditional probability density function; when M is discrete, it is the conditional probability mass function. If the mediator model directly estimates the conditional expectation ˆ µ M ( a, C M ) (e.g., M is binary or continuous, and some terms in the EIF are simplified for linearity in M ), then the parts of the formula involving ˆ f M are adjusted accordingly. For example, when M is binary (0 , 1) , ˆ f M ( M i | a, C M,i ) = ˆ µ M ( a, C M,i ) M i (1 -ˆ µ M ( a, C M,i )) 1 -M i .

Based on Equation (31), the influence functions for NDE and NIE are respectively:

In its implementation, the SIGMA framework computes the ψ NDE ( Z i ) and ψ NIE ( Z i ) scores for each sample according to the above structure, using nuisance function estimates ˆ η s obtained via cross-fitting.

A.3.5 Definition and Analysis of the Influence Function for Continuous Exposure in SIGMA

For mediation paths of length L ( π ) = 3 (i.e., A → M → Y ) involving a continuous exposure variable A , the SIGMA framework defines and utilizes the influence function described herein to estimate the Natural Direct Effect (NDE) and Natural Indirect Effect (NIE). This section details the mathematical construction of this influence function and the nuisance components it relies upon.

Nuisance Function Components for the AIF

The construction of this influence function is predicated on several key nuisance function components, estimated via K-Fold cross-fitting (as detailed in Section 4.4.1) for each (path π , DAG G s ) pair. Let Z i = ( A i , M i , Y i , C A,i , C M,i , C Y,i ) denote the data for the i -th observation, where C X,i represents the values of the parent set (covariates) of variable X under G s for sample i .

Conditional Expectation Model Predictions for Exposure A : The cross-fitted prediction for the conditional expectation of A given its covariates C A,i , obtained from a regression model (a multi-layer perceptron in SIGMA), is denoted as:

Global Estimate of Conditional Residual Variance of Exposure A : SIGMA assumes homoscedasticity for V ar ( A i | C A,i ) , positing V ar ( A i | C A,i ) = σ 2 A | C A ,res for all i . This global residual variance, denoted ˆ σ 2 A | C A ,res , is obtained by pooling the cross-fitted residuals e i = A i -ˆ µ A ( C A,i ) from all samples and calculating their sample variance (defined as 1 n ∑ n j =1 ( e j -¯ e ) 2 or 1 n -1 ∑ n j =1 ( e j -¯ e ) 2 , corresponding to the specific implementation).

Conditional Expectation Model Predictions for Outcome Y under Intervention on A with Observed M,C Y : For a given intervention level a on A (e.g., ˜ µ A or ˜ µ A + k ˜ σ A as defined in the main text), the cross-fitted prediction of the conditional expectation ˆ E [ Y i | A i = a, M obs i , C Y,i ] is denoted as:

:

Individualized Plugin Predictions and Sample Means of Target Counterfactual Expectations For any counterfactual expectation composing NDE and NIE, such as E [ Y ( a ′ , M ( a ′′ ))] (denoted γ a ′ a ′′ ):

The plugin prediction of the counterfactual outcome Y i ( a ′ , M i ( a ′′ )) for sample i is denoted as ˆ γ a ′ a ′′ ,i . This prediction is obtained via G-computation, recursively combining the nuisance functions (1)(3) above and outputs from the mediator model M M | A,C M (i.e., ˆ E [ M | A = a ′′ , C M ] or ˆ P ( M = m | A = a ′′ , C M ) ).And the sample mean of these individualized plugin predictions is denoted as ˆ γ a ′ a ′′ = 1 n ∑ j ˆ γ a ′ a ′′ ,j .

Here, ( a ′ , a ′′ ) represent specific exposure intervention level combinations required for defining NDE or NIE.

Mathematical Definition of the Influence Function

Within SIGMA, the influence function score ψ γ aa ∗ ( Z i ) for estimating a target counterfactual expectation (e.g., γ aa ∗ = E [ Y ( a, M ( a ∗ )) ) is defined as (corresponding to Equation (6) in the main text):

This influence function comprises two main parts:

textbfFirst Part (Exposure Model Correction Term): A i -ˆ µ A ( C A,i ) ˆ σ 2 A | C A ,res ( ˆ µ ( i ) Y | A = a,M obs ,C Y -ˆ γ aa ∗ ) . The term A i -ˆ µ A ( C A,i ) ˆ σ 2 A | C A ,res is the score function of the log-conditional likelihood of A | C A with respect to its mean parameter ˆ µ A ( C A,i ) , under the assumption that A | C A follows a Normal distribution N (ˆ µ A ( C A,i ) , ˆ σ 2 A | C A ,res ) . This score function is multiplied by a term related to the outcome model, (ˆ µ ( i ) Y | A = a,M obs ,C Y -ˆ γ aa ∗ ) , which reflects the deviation of the conditional outcome expectation (under intervened A = a , observed M,C Y ) from the mean target counterfactual expectation.

Second Part (Centered Plugin Term) : (ˆ γ aa ∗ ,i -ˆ γ aa ∗ ) . This term is the centralization of the individualized plugin estimate ˆ γ aa ∗ ,i of the target parameter with respect to its sample mean ˆ γ aa ∗ .

The overall influence functions for NDE and NIE are then derived from linear combinations of these influence function scores for their constituent counterfactual expectations (i.e., γ 10 , γ 00 , γ 11 , where the numerals 0 and 1 represent the reference level a ∗ and target level a , respectively), according to the definitions of NDE and NIE (NDE = γ 10 -γ 00 , NIE = γ 11 -γ 10 ). Specifically:

In these combinations, each ψ γ a ′ a ′′ ( Z i ; ˆ η s ) term on the right-hand side adopts the structure of Equation (34), utilizing the specific nuisance function components corresponding to its target counterfactual expectation ( γ 10 , γ 00 , or γ 11 ).

Construction Rationale and Discussion

The influence function for continuous exposure defined within the SIGMA framework (Equation (34)) is motivated by the goal of providing an estimation method that offers first-order bias correction over simple plugin estimators while maintaining computational tractability. Its structure incorporates elements common in influence function theory, such as leveraging score function information from the exposure model and centering plugin estimates.

The definition of this influence function does not explicitly include direct correction terms related to the mediator model P ( M | A,C M ) beyond its implicit role in the G-computation of ˆ γ aa ∗ ,i . In semiparametric theory, fully efficient influence functions for mediation effects often contain more complex terms explicitly dependent on the parameters or score functions of the mediator model(s) to ensure properties like double robustness against misspecification of either the outcome or mediator models. The influence function employed by SIGMA, while incorporating corrections related to the exposure model and centering the outcome predictions, may differ in its statistical properties, particularly double robustness and semiparametric efficiency, compared to such fully specified EIFs. Its performance is thus more reliant on the accurate estimation of all nuisance models, especially the conditional mean of the exposure ˆ µ A ( C A ) and the conditional expectations related to the outcome ˆ µ Y | A = a,M obs ,C Y . The adoption of a global homoscedastic approximation for the conditional variance of the exposure, ˆ σ 2 A | C A ,res , is a choice made for implementational simplicity.

Despite these specific implementational choices, by incorporating corrections related to the exposure model, this influence function is expected to offer more robust performance in finite samples compared to uncorrected plugin estimators, particularly when the exposure model is well-estimated. A detailed theoretical analysis of this influence function under various data-generating mechanisms and an in-depth study of its finite-sample behavior constitute important directions for future research.

B VAE-based Imputation Module

To address missing data, SG-RMA incorporates a Variational Autoencoder (VAE) specifically adapted for imputing mixed-type tabular data. Formally, we consider the raw data matrix X ∈ R N × P , which may contain missing entries, along with a binary mask matrix M ∈ { 0 , 1 } N × P indicating observed ( M ij = 0 ) or missing ( M ij = 1 ) values.

Data Preprocessing. Raw features are preprocessed according to their data types: continuous variables undergo standardization, whereas categorical variables are transformed via one-hot encoding,

expanding the feature dimensionality from P to P ′ . The resulting processed dataset is denoted as X ′ ∈ R N × P ′ . Missing entries within X ′ are initially filled with zeros to facilitate V AE training, and the corresponding mask M is concatenated to guide the imputation process explicitly.

VAE Model Specification. The VAE consists of an encoder network q ϕ ( z | X ′ , M ) and a decoder network p θ ( X ′ | z ) . Given an input vector ( X ′ i , M i ) , the encoder produces parameters of a Gaussian posterior distribution in the latent space: mean vector µ z i and variance vector σ 2 z i . The latent representation z i is then sampled via the reparameterization trick z i = µ z i + σ z i ⊙ ϵ , where ϵ ∼ N (0 , I ) . The decoder reconstructs data from latent representations, employing distinct output layers for each original feature type: linear layers for continuous features, logistic sigmoid activation for binary features, and softmax logits for categorical features.

The VAE optimization objective maximizes the evidence lower bound (ELBO), defined explicitly for observed entries as:

where x ′ i, obs denotes the observed elements of x ′ i . The reconstruction likelihood log p θ ( x ′ i, obs | z i ) aggregates squared errors for continuous features, binary cross-entropy losses for binary features, and categorical cross-entropy losses for categorical features. The prior distribution p ( z ) is standard normal, and β is a hyperparameter balancing reconstruction quality and latent regularization. Training employs the AdamW optimizer, adaptive learning rate schedules, and early stopping criteria based on validation performance.

Imputation Procedure. After model training, missing data imputation proceeds as follows: For each sample i , the trained encoder maps the filled input ( X ′ filled ,i , M i ) to the posterior mean latent representation µ z i . Passing this latent vector through the decoder yields reconstructed outputs ˆ x ′ i . Reconstruction outputs are then transformed back into the original data space, applying inverse scaling to continuous variables, thresholding via logistic sigmoid for binary variables, and category assignment via argmax operation for categorical variables. Finally, missing entries within the original dataset X are replaced by their respective reconstructions, while observed values remain unchanged. The imputation procedure is summarized in Algorithm 1.

B.1 Path Stability Score: Theoretical Properties

B.1.1 PSS as Posterior Probability Estimator

Section 4.3 of the main text introduces the Path Stability Score (PSS ( π ) ) as a measure to quantify the evidential support for a mediation pathway π . It is defined as PSS ( π ) = 1 N valid ∑ N valid s =1 I ( π ⊆ G s ) , where { G s } N valid s =1 is an ensemble of N valid valid Directed Acyclic Graphs (DAGs) sampled based on the output of the Flow-SEM stage (Section 4.2.3). This section provides a rigorous justification for the interpretation of PSS ( π ) as a Monte Carlo estimate of P ( π ⊆ G | D ) , the posterior probability that path π is present in a DAG G sampled from the (approximate) posterior distribution over graph structures learned by SIGMA given data D .

Let P ( G | D ) denote this SIGMA-approximated posterior distribution over the space of DAGs G , derived from the Flow-SEM model and the subsequent DAG sampling mechanism described in Section 4.2.3. The ensemble { G s } N valid s =1 consists of N valid valid DAGs treated as independent and identically distributed (i.i.d.) samples from P ( G | D ) .

Consider a specific path π . We define an indicator random variable X π ( G ) = I ( π ⊆ G ) , which equals 1 if path π is a subgraph of G , and 0 otherwise. The posterior probability of path π under P ( G | D ) is the expectation of this indicator variable:

The Path Stability Score, PSS ( π ) = 1 N valid ∑ N valid s =1 X π ( G s ) , is the sample mean of X π ( G ) based on these N valid i.i.d. samples.

Theorem 1 (Statistical Properties of PSS) . The Path Stability Score PSS ( π ) is:

1. An unbiased estimator of P ( π ⊆ G | D ) , i.e., E [ PSS ( π )] = P ( π ⊆ G | D ) .

Algorithm 1 VAE-based Missing Value Imputation for Mixed-Type Data

Require: Trained VAE (encoder q ϕ , decoder p θ ), Dataset X ∈ R N × P with missing values, Feature types { T j } P j =1 where T j ∈ { continuous , binary , categorical } , Preprocessing parameters (scaling factors { µ j , σ j } for continuous features, one-hot encoders for categorical features).

Ensure: Fully imputed dataset X ∈ R N × P .

A consistent estimator of P ( π ⊆ G | D ) , i.e., PSS ( π ) p - → P ( π ⊆ G | D ) as N valid →∞ .

Proof. (1) Unbiasedness: Let { G s } N valid s =1 be the set of N valid valid DAGs sampled i.i.d. from P ( G | D ) .

Since each G s ∼ P ( G | D ) (i.i.d.), E [ X π ( G s )] = E G ∼ P ( G | D ) [ X π ( G )] = P ( π ⊆ G | D ) . Thus,

(2) Consistency: The variables X π ( G s ) are i.i.d. Bernoulli random variables with mean p = P ( π ⊆ G | D ) and finite variance p (1 -p ) . By the Weak Law of Large Numbers (and Strong Law for almost sure convergence), their sample mean PSS ( π ) converges in probability to p as N valid →∞ .

Therefore, PSS ( π ) provides a statistically sound Monte Carlo approximation of the posterior probability of path π under the posterior distribution P ( G | D ) approximated by SIGMA. The accuracy of

this approximation depends on two primary factors: (i) the quality of P ( G | D ) as determined by the Flow-SEM and the DAG sampling mechanism, and (ii) the number of valid DAGs N valid .

B.1.2 Asymptotic Properties of PSS: Convergence Analysis with Increasing Sample Size

Theorem 1 establishes that the Path Stability Score, PSS ( π ) , serves as an unbiased and consistent estimator for p π ≡ P ( π ⊆ G | D ) . The consistency, derived from the Law of Large Numbers, ensures that with a sufficiently large ensemble of N valid valid DAGs, PSS ( π ) converges to p π . This section further elaborates on the asymptotic properties of PSS ( π ) as N valid increases, particularly its rate of convergence and distributional characteristics, which are fundamental for understanding its reliability and for guiding the choice of N valid .

Asymptotic Normality via the Central Limit Theorem (CLT). Since PSS ( π ) is a sample mean of N valid independent and identically distributed (i.i.d.) Bernoulli random variables X π ( G s ) = I ( π ⊆ G s ) (each with mean p π and finite variance p π (1 -p π ) ), the Central Limit Theorem (CLT) applies. The CLT states that for a sufficiently large N valid :

where d - → denotes convergence in distribution. Equivalently, for large N valid, PSS ( π ) itself is approximately normally distributed:

Variance Analysis and Rate of Convergence. The exact variance of the PSS ( π ) estimator is:

This shows that the variance is inversely proportional to N valid . The standard error of PSS ( π ) is thus SE ( PSS ( π )) = √ p π (1 -p π ) /N valid, decreasing at a rate of O (1 / √ N valid ) . This implies that to halve the standard error, N valid must be quadrupled. The variance is maximized when p π = 0 . 5 , indicating that more samples are needed for precise estimation when the true path probability is near 0 . 5 .

Confidence Interval Construction and Practical Considerations for N valid . The asymptotic normality (Equation 40) allows for the construction of approximate Wald-type confidence intervals for p π :

where z 1 -α/ 2 is the (1 -α/ 2) -quantile of the standard normal distribution, and PSS ( π ) is used to estimate p π in the standard error term. This interval provides a range of plausible values for the true posterior probability p π . The reliability of this interval and the PSS estimate itself depends on N valid . For probabilities p π very close to 0 or 1 , fewer samples may suffice for a given precision.

Connection to Bayesian Model Averaging (BMA). The statistical properties of PSS, particularly its interpretation as a posterior probability estimate, directly support the subsequent Bayesian Model Averaging (BMA) stage of SIGMA (Section 4.4.3). By using a threshold τ on PSS ( π ) to identify stable paths, SIGMA focuses computational resources on pathways with substantial evidential support under P ( G | D ) . The ensemble { G s } N valid s =1 used to compute PSS is the same basis for BMA, ensuring a coherent propagation of structural uncertainty. For a stable path π , the BMA estimate ˆ θ BMA ( π ) and its variance ˆ V ( ˆ θ BMA ( π )) represent the model-averaged effect and its associated uncertainty. This averaging is performed over the plausible graph structures within the ensemble that support path π , effectively integrating over the structural uncertainty concerning these specific pathways as captured by P ( G | D ) .

B.2 Discussion on path stability score threshold

The Path Stability Score (PSS) is defined as the frequency of path π in the posterior DAG sample set:

where π represents the candidate mediational path, G s is the s -th DAG sample drawn from the structural posterior distribution, N is the total number of DAG samples, and I ( · ) is the indicator function. Given a threshold τ ∈ [0 , 1] for the path stability score, we define the stable path set as:

The selection of threshold τ directly determines the size and composition of Π stable, thereby further influencing the Bayesian Model Averaging (BMA) estimate of path-specific effects:

where G π ( τ ) = { G s : π ⊆ G s , PSS ( π ) ≥ τ } represents the subset of DAGs containing path π given threshold τ , and ˆ θ s ( π ) is the mediation effect estimated based on the DAG sample G s .

A smaller value of τ may lead to the inclusion of more low-frequency paths, increasing the risk of false path discoveries (false positives); a larger value of τ may exclude paths with real effects but weaker statistical support (increasing false negatives). Therefore, in practice, the recommended strategy is to select a moderate intermediate value (e.g., τ = 0 . 15 ), and then make fine adjustments based on the specific characteristics of the data and prior knowledge in the relevant domain. We explicitly adopted this strategy in the numerical experiments and practical algorithm implementation in this paper, and observed that this approach can achieve a reasonable balance between robustness and sensitivity in path selection.

C Related Work

Causal Structure Learning. Differentiable neural approaches have advanced causal DAG discovery by casting it as continuous optimization. Zheng et al. (2018) introduced the seminal NOTEARS [72] method, which enforces an analytic acyclicity constraint to learn DAGs via standard gradient-based solvers. This breakthrough spurred many extensions to capture nonlinear relations and improve scalability. For example, GraN-DAG [35] extended NOTEARS to nonlinear settings by using feedforward networks to model causal mechanisms, and DAG-GNN [69] embedded a variational autoencoder with a graph neural network to learn DAG structures from data. Subsequent works proposed alternative differentiable formulations of the acyclicity constraint for stability and speed. ENCO [38] optimizes independent edge likelihoods without explicit acyclicity constraints, while Nazaret et al. introduce a spectral constraint in a Stable DCD [44] approach to improve numerical stability and scale causal discovery to thousands of variables .

Reinforcement learning (RL) has also been applied to structure learning - e.g., Zhu and Chen [75] use an RL agent to sequentially build the DAG, avoiding expensive combinatorial searches. Beyond point estimation of a single graph, recent variational methods estimate a distribution over plausible DAGs. For instance, BCD-Nets [16] perform Bayesian causal discovery via variational inference, yielding a posterior over DAG structures and quantifying uncertainty. Similarly, Differentiable Causal Discovery with Interventions ( DCDI ) [12] integrates normalizing flows to model complex causal mechanisms under interventions.

Mediation Analysis. Deep learning has also been applied to mediation analysis to estimate pathspecific effects (natural direct and indirect effects) in complex settings. DeepMed [65] is a recent example that uses deep neural networks to flexibly estimate the necessary nuisance functions in the efficient influence function for mediation, achieving semiparametric-optimal estimation of direct and indirect effects. By cross-fitting DNNs for propensity and outcome models, DeepMed debiases mediator and outcome predictions and attains the efficiency bound without restrictive model assumptions. Other works leverage deep latent-variable models to handle mediators and hidden confounding.

CMA-VAE [13] introduced a variational autoencoder for causal mediation that accounts for latent confounders, enabling estimation of natural direct and indirect effects even with unobserved confounders. Building on this, Disentangled Mediation Analysis VAE (DMAVAE) [66] separates latent factors into mediating, confounding, and irrelevant components. This disentanglement allows DMAVAE to consistently estimate direct and indirect effects under weaker assumptions than traditional sequential ignorability, significantly improving mediation effect estimation in the presence of multiple types of latent confounders.

In parallel, deep models have been used for path-specific effect estimation in high-dimensional settings like text and fairness. CausalBERT [63] fine-tunes transformer language models to learn low-dimensional text embeddings that retain information needed for causal adjustment. This approach enables estimation of causal effects from text (e.g., the effect of a review's content on outcomes) by adjusting for textual confounders within the embedding space. Similarly, in algorithmic fairness, researchers have examined path-specific effects via deep nets to detect mediated discrimination. For example, Chiappa [14] studies counterfactual fairness by removing information about protected attributes along certain causal paths, and Nabi & Shpitser [43] develop methods to identify pathspecific effects under mediation and confounding. These efforts underscore a growing theme: integrating deep representation learning with causal mediation analysis to uncover nuanced causal pathways (e.g., via stability analysis or regularization to select stable mediators) in complex, highdimensional data.

Deep Treatment Effect Estimation. Estimating individualized treatment effects (ITEs) or heterogeneous causal effects has benefited greatly from deep learning, particularly through representation learning and efficient influence function (EIF) techniques. A landmark work by Johansson et al. [31] introduced learning balanced representations for treatment and control groups to improve ITE generalization. This approach, extended by Shalit et al. [53] with integral probability metric regularization, underlies methods like TARNet/CFR that reduce selection bias by aligning latent representations across treatment groups. Subsequent methods added adversarial objectives to enforce balance: Yao et al. [67] propose a SITE approach that preserves local similarity while using adversarial training to minimize covariate distribution differences, and GANITE [68] directly employs GANs to generate counterfactual outcomes for ITE estimation.

Another line of work incorporates semiparametric theory to improve estimation efficiency and robustness. DragonNet [54] is a neural architecture that adapts the network training to target the efficient influence function of the average treatment effect. By jointly learning the outcome and propensity predictor with a targeted regularization term, DragonNet yields doubly-robust estimates that empirically outperform earlier architectures. In a related vein, CEV AE [39] uses deep variational inference to handle unobserved confounders, learning a latent variable model that estimates treatment effects under a causal graphical model assumption. Extensions of CEVAE have further improved disentanglement; for instance, TEDVAE [70] factorizes latent confounders into multiple parts to better separate causal effects.

To inject inductive biases about potential outcomes, FlexTENet [17] adaptively learns what representations to share between the treated and control outcome functions, reflecting the assumption that many predictive factors are common between potential outcomes. Beyond standard observational studies, deep models have tackled more complex causal inference tasks: DeepIV [25] uses deep nets for instrumental variable regression to estimate counterfactual predictions in the presence of hidden confounding, and other works address time-series and panel data with recurrent networks for sequential treatment effects [5].

Causal Inference under Structural Uncertainty A distinguishing challenge for methods like SIGMA is causal inference under structural uncertainty , where the true causal graph is unknown or ambiguous. Recent research has begun to explicitly account for this uncertainty using Bayesian deep learning, model averaging, and ensembles of causal models. One approach is to perform Bayesian model averaging (BMA) over DAGs: rather than committing to a single estimated graph, one can weight effect estimates by each graph's posterior probability. Cundy et al. [16] implement this via their variational BCD-Nets, which output a distribution over possible DAG structures. By sampling DAGs from the learned posterior, BCD-Nets can compute expectation of causal effects across many graph realizations, inherently providing credible intervals for effects. DECI [22], a deep end-to-end causal inference framework, uses normalizing flows within a variational Bayesian scheme.

DECI simultaneously learns a DAG posterior and estimates causal quantities, essentially performing causal discovery and inference in one unified model.

Furthermore, techniques like stability selection have been adapted to causal structure learning to identify consensus edges under data perturbations, thereby informing model averaging strategies (e.g., selecting only edges that consistently appear across high-scoring DAGs) [44]. Such ideas align with the stability analysis component of SIGMA, which seeks robust mediation pathways by exploring variations in the learned structure. Overall, the literature suggests that accounting for structural uncertainty - through Bayesian posterior averaging, DAG ensembles, or stability-based model selection - can substantially improve the reliability of causal effect estimates in complex models. SIGMA builds on these insights by integrating BMA of mediation pathways with EIF-based effect estimation, ensuring that uncertainty in the causal graph is properly propagated into uncertainty in estimated direct and indirect effects.

D Human Phenotype Project Dataset

D.1 Description of HPP Cohort

The Human Phenotype Project (HPP) is a deeply phenotyped, longitudinal cohort study conducted in Israel, designed to investigate the interplay between various physiological systems and environmental exposures. The present study analyzed data collected from 6,748 participants aged 40-75 years between January 2019 and December 2022. All participants provided written informed consent, and the study protocol was approved by the institutional review board of the Weizmann Institute of Science (IRB no. 1719-1). Participants underwent comprehensive profiling, including clinical, physiological, behavioral, and multi-omics assessments, which were categorized into 17 body systems [34], one of which focused on sleep. Sleep characteristics were derived from home sleep apnea testing (HSAT) using the WatchPAT 300 device (ZOLL Itamar), with each participant completing up to two monitoring series, each comprising three nights of testing over a two-week period. A total of 16,812 nights of valid HSAT recordings from 6,366 individuals (47.8% male, 52.2% female; mean age 52.4 ± 7.7 years; mean BMI 26.1 ± 4.1 kg/m 2 ) were included in the final analysis.

The cohort primarily consisted of healthy, educated individuals of European (Ashkenazi) Jewish ancestry residing in a relatively homogenous environmental context. Exclusion criteria included severe medical conditions at enrollment. Participants were scheduled for biennial follow-up assessments over a 25-year period. This extensive data resource enables a high-resolution investigation of associations between sleep and systemic health.

D.2 Body System-Derived Features

We utilize data from 11 body systems, comprising over 500 raw clinical and physiological attributes. To facilitate tractable and clinically meaningful mediation analysis, we derived a set of 62 aggregated features by domain-informed grouping of related variables across systems. These features are constructed to capture coherent physiological constructs suitable for pathway-level causal interpretation. Please see Table 3 for the detailed description of our aggregated features.

Sleep Characteristic Sleep characteristics were captured using the WatchPAT 300 home sleep monitoring device across up to three nights per participant. A total of 448 features were extracted, comprising two main subgroups: 100 sleep test measurements and 348 pulse rate variability (PRV) features. Sleep test features included metrics related to respiratory events (e.g., peripheral apnea-hypopnea index [pAHI]), oxygen desaturation (e.g., mean SpO2 nadir, time below 90% saturation), snoring intensity, sleep stage distribution (light, deep, REM), sleep efficiency, and body position during sleep. PRV features were derived from the peripheral arterial tonometry signal using the NeuroKit2 library [42], spanning time- and frequency-domain indices, entropy, and nonlinear dynamics, computed across REM, NREM, wake, and full-night segments.

Metabolic Pathways Metabolic pathway features were derived from metagenomic profiling of the gut microbiome using the HUMAnN3 [4] functional annotation pipeline. Relative abundances of microbial metabolic pathways were quantified based on high-throughput sequencing data, followed by taxonomic and functional mapping against a curated reference database. Only pathways present in

at least 5 % of participants were retained for analysis, and abundance values were log10-transformed to stabilize variance. These pathways represent a functional summary of microbial activity, covering key domains such as amino acid biosynthesis, vitamin and cofactor metabolism, carbohydrate degradation, and short-chain fatty acid production. They were used to probe host-microbiome interactions relevant to systemic metabolic health and potential associations with sleep-disordered breathing phenotypes.

Cardiovascular System Cardiovascular features encompassed a comprehensive set of hemodynamic and structural measurements obtained through non-invasive techniques. Peripheral and central

blood pressures were measured in sitting, supine, and standing positions. Arterial stiffness was evaluated using pulse wave velocity (PWV) recorded with the Falcon device (Viasonix), and endothelial function was indirectly assessed through carotid intima-media thickness using high-resolution ultrasonography (SuperSonic Aixplorer MACH 30). Retinal microvascular parameters were extracted via fundus imaging (iCare DRSplus) and processed with the AutoMorph [74] Python package. Cardiac electrical activity was characterized by 12-lead resting ECG (PC-ECG 1200, NORAV), from which standard and derived indices (e.g., heart rate, QT intervals) were calculated. The cardiovascular dataset thus integrates structural, functional, and electrophysiological dimensions of cardiovascular health, enabling multiscale association testing with sleep traits.

Inflammation The inflammation system features were derived from standard clinical laboratory tests obtained from participants' health maintenance organizations (HMOs) as part of their routine medical care. Participants were encouraged to upload results from various tests, including those indicative of systemic inflammation. Key biomarkers relevant to the inflammation system included C-reactive protein (CRP), white blood cell (WBC) count, and differential counts such as neutrophils, lymphocytes, monocytes, eosinophils, and basophils. These markers provide insights into both acute and chronic inflammatory states. CRP, an acute-phase protein synthesized by the liver, serves as a sensitive indicator of systemic inflammation and is commonly elevated in response to infection or tissue injury. The WBC count and its differentials offer information on the immune system's cellular components, with variations potentially reflecting underlying inflammatory or immune responses. All laboratory values were standardized and stored in a structured format to facilitate downstream analyses.

Baseline Characteristics (BMI) Body mass index (BMI) was calculated as weight in kilograms divided by height in meters squared (kg/m 2 ) and was treated as a baseline covariate alongside age and sex. Though BMI is not classified as a standalone body system in this analysis, it serves as a critical covariate due to its known influence on both sleep-disordered breathing and cardiometabolic risk. It was excluded from the body composition system to avoid redundancy in statistical modeling.

E Synthetic Datasets Details

E.1 Generation Process

Weconstruct synthetic datasets using a modular simulation framework that mimics realistic causal systems with configurable complexity. The generation process is parameterized to control graph structure, variable types, functional complexity, effect magnitudes, missing data patterns, and more, making the synthetic data suitable for benchmarking causal discovery and mediation analysis algorithms. The entire dataset generation procedure is outlined in Algorithm 2.

Causal Graph Construction Each dataset is generated from a DAG G over p variables and contains n observational samples. The graph is sampled from an Erd˝ os-Rényi (ER) model with a specified expected average in-degree k , followed by a random topological sort to ensure acyclicity. The DAG defines the full set of causal relationships among the p variables, with no predefined roles assigned to individual variables. To support the construction of realistic and diverse causal structures, we explicitly control two key properties of the graph: the chain path length ℓ , which determines the depth of the longest directed chain in the DAG, and the number of structurally parallel paths κ , which specifies how many disjoint or partially overlapping causal routes exist across the graph. This parameterization allows us to generate graphs that reflect varying levels of structural complexity, enabling robust evaluation of causal discovery and inference methods under different topological regimes.

Structural Equation Modeling Let v = ( v 1 , · · · , v p ) be a collection of random variables. Each node is randomly assigned a variable type according to a multinomial distribution over three categories: continuous with probability π c , binary with π b , and categorical (with up to L cat levels) with probability π cat, where π c + π b + π cat = 1 . Once the types are assigned, we generate variables according to a structural equation modeling (SEM) [46] framework based on a given DAG. Specifically, each variable v i is assigned a structural equation of the form:

where z i is an exogenous noise term independent of all other variables, pa ( i ; G ) denotes the set of parent nodes of i in the graph G , and F i is the deterministic function governing the influence of parent variables on v i . In this work, we focus on a special class of SEMs known as additive noise models (ANMs), where the structural equations take the simplified form:

The functional form f i may be linear or nonlinear, including polynomial (e.g., v 2 ), sinusoidal (e.g., sin( v ) ), or exponential (e.g., exp( -| v | ) ) transformations. To control the proportion of nonlinear mechanisms, we define a nonlinearity ratio parameter ρ nonlin ∈ [0 , 1] , which specifies the expected fraction of nodes governed by nonlinear functions. For nodes not selected as nonlinear, f i defaults to a linear combination of parent variables. This stochastic selection enables systematic benchmarking across datasets with different levels of structural complexity. Pairwise interactions between parent variables can also be incorporated to increase structural complexity. The exogenous noise z i is sampled from a mixture of distributions-Gaussian, exponential, or Studentt -with user-defined mixture weights to simulate heteroskedasticity and non-Gaussian variability. For discrete variables, logistic or softmax functions are used to generate probabilistic outcomes conditioned on parent variables.

Causal Effect Scaling To enable controlled experimentation with mediation analysis, the generator includes an effect-scaling mechanism that adjusts the magnitude of causal effects along specific paths. Structural coefficients on the direct path from treatment to outcome, and on the indirect paths through mediators, are scaled such that the resulting absolute values of the natural direct effect (NDE) and natural indirect effect (NIE) fall within user-specified intervals, denoted by [ τ min NDE , τ max NDE ] and [ τ min NIE , τ max NIE ] , respectively. This ensures that generated datasets reflect a wide spectrum of causal regimes, from dominantly direct to heavily mediated effects.

Missingness Injection and Ground Truth Estimation To further enhance realism, the framework optionally injects missingness into the dataset under both missing completely at random (MCAR) and missing at random (MAR) mechanisms. The overall missingness rate r miss determines the proportion of missing entries. In addition, the framework allows configuration of the proportion of variables and the conditional dependencies under which missingness occurs. Ground-truth causal effects, including total effect (TE), NDE, and NIE, are estimated via Monte Carlo simulation using a large number of samples (e.g., 10,000 or more) by evaluating counterfactual outcomes under interventions on treatment and mediator variables.

E.2 Mediation Pathway Structures

To evaluate mediation analysis methods under diverse structural assumptions, we incorporate a range of causal graphs with varying mediation configurations into the synthetic data generator. Figure 4 illustrates several representative DAG structures that are automatically instantiated within our framework. These examples cover key topologies relevant for causal mediation, including both simple and complex pathways, as well as potential nonlinear effects.

X

Y

(a) Two node

X

1

X

2

M

Y

(b) Single Mediator with Confounding

X

1

X

2

M

1

M

2

Y

(c) Dual Independent Mediators

X

M

1

M

2

Y

(d) Nonlinear Dual Mediators

X

M

1

M

2

M

3

Y

(a) Direct-only structure ( 'Two-Node Path' ): This minimal structure includes only a direct effect from treatment X to outcome Y , with no mediators. It serves as a baseline for evaluating spurious mediation detection.

(b) Single mediator with confounding inputs ( 'Single Mediator with Confounding' ): A single mediator M lies between X and Y , but additional variables (e.g., X 1 , X 2 ) influence the mediator, simulating a confounded or high-dimensional mediator function.

(c) Parallel mediation ( 'Dual Independent Mediators' ): Two mediators M 1 and M 2 are both affected by X and influence Y independently, forming parallel mediation paths.

(d) Nonlinear parallel mediation ( 'Nonlinear Dual Mediators' ): Similar to (c), but the structural functions between nodes involve nonlinearities such as interactions or sine/exponential transforms.

e) Mixed mediation ( 'Hybrid Chain-and-Parallel Mediation' ): A more complex topology where two mediators M 1 and M 2 form parallel channels, and a third node M 3 forms a chain extending toward Y . This structure tests the ability of algorithms to disentangle indirect effects across multiple pathways.

E.3 Dataset Benchmarks

Table 4 summarizes the configurations used to generate representative datasets in our experiments. These settings allow us to benchmark the performance of causal inference methods across regimes with varying dimensionality, mediation complexity, and data quality. Specifically, we hold constant the overall variable type proportions ( π c = 0.6, π b = 0.3, π cat = 0.1), category levels ( L cat = 5), missingness ( r miss = 0.1), average in-degree ( k = 3) and the target standard deviation ranges of the causal effects ( τ NDE ∈ [0 . 2 , 0 . 4] , τ NIE ∈ [0 . 15 , 0 . 3] ). This design allows for controlled benchmarking across varying causal and statistical challenges.

To further validate the quality of mediation structures embedded in the synthetic datasets, we visualize the variable relationships of sampled mediation pathways. Specifically, we examine the empirical dependencies among treatment, mediator, and outcome variables to see whether the intended causal effects are faithfully reflected in the generated data. We analyze four types of associations: (i) Treatment → Mediator, (ii) Mediator → Outcome, (iii) Treatment → Outcome, and (iv) Mediator

→ Outcome under different treatment. These visualizations provide intuitive verification that the synthetic data preserves the expected mediation mechanisms and highlight the nonlinear or conditional dependencies that may arise under the configured data-generating process.

LowDim-L: LowDim-L serves as the baseline configuration in the benchmark suite, featuring low dimensionality ( p = 20 ), a fully linear structural specification ( ρ nonlin = 0 . 0 ), and a simple graph topology with a single short causal path ( κ = 1 , ℓ = 3 ). The graph structure is minimal and interpretable, offering a clean testbed for evaluating algorithmic correctness in effect estimation under idealized, statistically well-behaved conditions. The causal graph consists of a binary treatment variable V 8 , a continuous mediator V 2 , a categorical mediator V 18 , and a binary outcome V 13 . This setup enables clean interpretability of causal effects and serves as a reference for assessing performance under idealized conditions. LowDim-L provides a controlled environment for validating the correctness of pathway identification and effect estimation algorithms under structurally simple and statistically well-behaved conditions.

V8→V2

MediatorV2→OutcomeV13

MediatorV18→OutcomeV13

1.0

V18=0.0

1.0

Probability(V13=1)

1.0

V18=1.0

0.8

V18=2.0

80

0.8

P(V18| V8)

0.4

0.6

V18=3.0

V13

0.6

0.4

P(V13=1)

0.4

0.49

0.46

0.41

0.38

0.2

0.26 0.250.24 0.25

0.27

0.23 0.25 0.25

0.2

0.2

0

V8

0.0

V8=0.0

V8

V8=1.0

00

3

2

-1

0

V2

1

2

3

0.0

0.0

1.0

V18

2.0

3.0

V2→V13|V8

V18→V13|V8

V8→V13

1.0

Probability(V13=1)IV8=0.0

1.0

V8=0.0

1.0

V8=0.0

Probability(V13=1)/V8=1.0

V8=1.0

V8=1.0

MediationPathway

0.8

0.8

80

V13)

0.6

P(V13=1)

0.6

061

0.58

0.54

Treatment:V8;

0.4

0.4

0.4g0.51

0.44

0.47

0.38

0.45

0.36

0.40

0.42

0.46

0.2

0.2

0.2

Outcome: V13;

0.0

-2

V2

0.0

0.0

1.0

V18

2.0

3.0

0.0

0.0

V13

1.0

LowDim-N: LowDim-N mirrors the low-dimensional structure of LowDim-L ( p = 20 , n = 6000 ), but introduces nonlinear functional mechanisms ( ρ nonlin = 0 . 5 ) while maintaining a structurally simple graph with one short directed path ( κ = 1 , ℓ = 3 ). This setting incorporates nonlinear mappings within the graph's structural equations, allowing us to evaluate whether causal inference methods remain robust to nonlinearity. The variable types are consistent: treatment V 8 is binary, mediators V 2 and V 18 are continuous and categorical respectively, and outcome V 13 is binary. In contrast to LowDim-L , the functional dependencies here include non-additive effects and interactions, enabling the examination of empirical recoverability under nonlinear data-generating processes. In particular, the direction of the mediator-outcome relationship is reversed: V 2 now has a positive rather than negative effect on V 13, consistent with the design of its generating function. Conditional plots further highlight interactions modulated by treatment levels, providing visual confirmation of nonlinear mediation behavior.

V8→V2

MediatorV2→OutcomeV13

MediatorV18→OutcomeV13

8

1.0

V18=0.0

1.0

.0

V18=1.0

0.83

0.8

V18=2.0

80

0.8

0.78

2

P(V18| V8)

0.6

V18=3.0

V13

0.6

P(V13=1)

061

0.55

0.67

0.4

0.4

0.4

0.2

0.27

0.23 0.25 0.25

0.2

Probability(V13=1)

8

V8

1

0.0

V8=0.0

V8

V8=1.0

00

2

-1

0

V2

2

3

0.0

0.0

1.0

V18

2.0

3.0

V2→V13|V8

V18→V13|V8

V8→V13

1.0

1.0

V8=0.0

0.95

1.0

V8=0.0

V8=1.0

0.88

V8=1.0

MediationPathway

V13)

0.8

0.6

P(V13=1)

0.6

0.8

0.59

0.51

0.62

0.72

0.69

0.72

0.6

80

0.65

0.76

Treatment:V8;

0.4

0.4

0.35

0.2

Probability(V13=1/V8=0.0)

0.2

0.24

Outcome: V13;

Probability(V13=1/V8=1.0)

0.0

0

N

0.0

0.0

1.0

V18

2.0

3.0

0.0

0.0

V13

1.0

V2

LowDim-P: LowDim-P is designed to introduce structural complexity through multiple causal pathways, while maintaining low dimensionality ( p = 20 , n = 6000 ) and moderate nonlinearity ( ρ nonlin = 0 . 5 ). The graph contains κ = 2 parallel paths of moderate depth ( ℓ = 3 ), representing concurrent channels of influence. The variable configuration includes a binary treatment V8, continuous mediator V2, categorical mediator V18, binary mediator V7, and binary outcome V13. This setup is designed to evaluate the interaction and combined influence of multiple concurrent mediators under nonlinear conditions.

V8 →V2

V8 →V7

V2 →V13

1.0

V18=0.0

V8=0.0

1.0

Probability(V113=1)

V18=1.0

V8=1.0

0.8

V18=2.0

0.8

0.8

P(V18|V8)

V18=3.0

0.61

0.6

0.48

0.52

V13

0.39

0.26 0.25 0.24 0.25

0.27

0.230.250.25

0.2

20

0.2

。

0

V8

0.0

V8=0.0

V8

V8=1.0

0.0

V7

1.0

0.0

20

-15

-10

V2

-5

0

V18→V13

V7→V13

V2 →V13|V8

V18 → V13 | V8

1.0

V7=0.0

1.0

Probability(V13=1IV8=0.0)

V8=0.0

V7=1.0

Probability(V13=1|V8=1.0)

V8=1.0

8'0

0.8

0.8

0.8

0.64

0.67

0.6

0.6

V13

0.52

0.2

0.45

0.35

0.33

0.28

0.2

0.36

0.33

0.2

0.2

0.4

0.40

0.29

0.43

0.26

0.41

0.22

0.35

0.0

0.0

1.0

V18

2.0

3.0

0.0

0.0

V13

1.0

0.0

20

-15

0.0

V18

2.0

3.0

V7 → V13 | V8

V8→V13

V8=0.0

1.0

V8=0.0

V8=1.0

V8=1.0

MediationPathway

0.8

0.71

P(V13=1)

0.6

P(V13V8)

.6-

0.57

Treatment:V8;

0.4

0.44

0.42

0.4

0.43

Mediator:V2,V7,V18;

0.28

0.29

0.29

02

0.2

Outcome: V13;

0.0

V7=0.0

0=

0.0

0'0

1.0

V7

V13

LowDim-D: LowDim-D represents a more structurally complex synthetic setting with p = 20 , n = 6000 , moderate nonlinearity ( ρ nonlin = 0 . 5 ), and a long-chain causal structure characterized by a single deep path ( κ = 1 , ℓ = 6 ). It includes a binary treatment variable V 8 , continuous mediators V 2 , V 16 , and V 19 , a categorical mediator V 18 , a binary mediator V 11 , and a binary outcome V 13 . This configuration captures extended sequential dependencies, where indirect effects propagate through multiple intermediate variables. It provides a testbed for assessing robustness in effect estimation under deeper, multi-step causal pathways.

02

V8 →V16

V8 → V2

V8 →V19

1.0

V18=0.0

V18=1.0

0.8

V18=2.0

P(V18| V8)

V18=3.0

V16

V19

2

0.4

目

0.26 0.25 0.24 025

0.27

0.23 0.25 0.25

4

0.2

6

0

V8

0

V8

V8=0.0

V8

V8=1.0

0

V8

1

V8 →V11

V16→V13

V2 →V13

V18→V13

V8=0.0

1.0

1.0

10

V8=1.0

0.8

0.75

0.76

0.8

0.8

0.8

0.72

0.72

0.62

0.65

V13

0.6

V13

0.4

0.2

0.25

0.24

0.2

0.2

0.2

Probability(V13=1)

Probability(V13=1)

0.0

0.0

0.0

V11

1.0

5

V16

10

6

-4

V2

0

2

0.0

1.0

V18

20

3.0

V19→V13

V11→V13

V16→V13|V8

V2 → V13 | V8

1.0

1.0

V11=0.0

1.0

10

V11=1.0

0.8

80

0.74

0.8

0.8

0.6

P(V13|V11)

0.6-

0.53

0.47

V13

V13

0.4

.4-

0.4

0.4

0.26

Probability(V13=1|V8=0.0)

0.2

Probability(V13=1|V8=0.0)

Probability(V13=1)

Probability(V13=1/V8=1.0)

Probability(V13=1V8=1.0)

-6

-4

V19

0

2

0.0

0.0

V13

1.0

0.0

-5

0

V16

5

10

15

0.0

6

-4

V2

0

V18→V13| V8

V19 →V13| V8

V11 →V13 |V8

V8→V13

V8=0.0

1.0

1.0

V8=0.0

1.0

V8=0.0

V8=1.0

V8=1.0

V8=1.0

P(V13=1)

0.8

0.63 0.62

0.65 0.65

0.72 0.72

0.72 0.73

V13

0.6

0.8

P(V13=1)

0.8

0.74

0.74

0.8

0.68

0.68

0.49

0.45

0.4

0.4

0.4

0.32

0.32

0.2

Probability(V13=1/V8=0.0)

0.2

0.2

Probability(V13=1|V8=1.0)

0.0

0.0

V18

2.0

3.0

0.0

V19

-2

0.0

0.0

V11

0.0

0.0

V13

1.0

MidDim-S: MidDim-S increases both dimensionality and structural complexity, with p = 100 , n = 6000 , and a moderate level of nonlinearity ( ρ nonlin = 0 . 5 ), while preserving a simple causal topology with a single short directed path ( κ = 1 , ℓ = 3 ). The treatment variable V 35 is binary, mediators include a categorical variable V 2 , a binary variable V 13 , and a continuous variable V 14 , with V 17 serving as the continuous outcome. This setup is designed to assess algorithm performance in mid-dimensional regimes where the causal structure remains sparse and shallow, isolating the effect of dimensional scaling under otherwise controlled conditions.

V35→V2

V35→V13

V35→V14

V2→V17

10

V2=0.0

V35=0.0

1.0

V2=1.0

V35=1.0

0.5

0.8

V2=2.0

0.8

P(V2| V35)

V2=3.0

P(V13/V35)

0.62

0.0

0.6

0.52

0.48

.0

V17

.5-

0.4

0.38

0.5

1.0

0.25 0.25 024 0.26

0.25 0.25 0.25 0.25

-1.5

0.2

0.2

0-

-2.0

8.

1.5

2.5

O

V35=0.0

V35

V35=1.0

0.0

V13

1.0

V35

。

V2

1

V13→V17

V14→V17

V2 →V17 | V35

V13 →V17| V35

V35=0.0

V35=0.0

V35=1.0

录：

V35=1.0

V17

。

。

1.0

0.5

0.0

0.5

1.0

V2=0.0

V2=1.0

V2=2.0

V2=3.0

V130.0

V131.0

V13

V14

V2

V13

V14 →V17|V35

V35→V17

MediationPathway

Treatment:V35;

V17

V17

Mediator:V2,V13,V14;

V35=0.0

Outcome:V17;

V35=1.0

8

-1.0

0.5

0.0

0.5

1.0

V14

V35

MidDim-D: MidDim-D extends the structural complexity of the mid-dimensional setting with p = 100 , n = 6000 , a nonlinearity ratio of ρ nonlin = 0 . 5 , and a long-chain causal configuration characterized by a single extended path ( κ = 1 , ℓ = 6 ). The DAG includes multiple layers of intermediate variables connected through a deep sequential structure, challenging algorithms to recover indirect effects that accumulate across heterogeneous transformations. The treatment variable V 35 is binary, and the mediation pathway involves heterogeneous mediators: a categorical variable V 2 , binary variables V 13, V 15 and V 89, and continuous variables V 14 and V 40, with a continuous outcome V 17 . This dataset is well-suited for evaluating a method's ability to capture layered, nonlinear, and high-dimensional mediation dynamics.

V35 →V2

V35→V40

V35→V89

V35→V13

1.0

V2=0.0

V35=0.0

1.0

V35=0.0

V2=1.0

V35=1.0

V35=1.0

0.8

V2=2.0

0.8

0.8

0.71

P(V2| V35)

V2=3.0

P(V89|V35)

0.62

0.61

P(V13|V35)

oot

0.62

V40

0.4

0.38

0.39

0.38

0.250.250.240.26

0.25 0.25 0.25 0.25

0.29

0.2

0.2

0.0

V35=0.0

V35

V35=1.0

V35

0'0

0.0

V89

1.0

0.0

0.0

V13

1.0

V35→V14

V35→V15

V2→V17

V40→V17

8

1.0

V35=0.0

0

V35=1.0

o

0.8

0.74

8

P(V15/V35)

V14

0.6

V17

V17

04

0.26

0.21

0

。

F

0

V35

0.0

0.0

V15

1.0

V2

-1

V40

2

V89→V17

V13→V17

V14 →V17

V15→V17

。

亚

V17

-1

V89

V13

V14

0

0

V15

V2→V17|V35

V40→V17|V35

V89→V17|V35

V13→V17|V35

■

■

V35=1.0

V35=0.0

。

8

■

■

V35=0.0

V35=1.0

8

■

V35=0.0

V35=1.0

■

V35=0.0

V35=1.0

通

f

F

室

V17

V17

V17

V17

V2=0.0

V2=1.0

V2

V2=2.0

V2=3.0

2

-1

V40

2

V89=0.0

V89

V89*1.0

V130.0

V13

V13=1.0

V14 →V17|V35

V15→V17|V35

V35→V17

V35=0.0

V35=1.0

8

。

MediationPathway

业

Treatment:V35;

V17

V17

V17

Mediator:V2,V13,V14,

V15,V40,V89;

■

V35=0.0

Outcome: V17;

V35=1.0

-2

V15=0.0

V15=1.0

V14

V15

V35

HigDim-S: HighDim-S explores a high-dimensional setting with p = 200 , n = 6000 , moderate nonlinearity ( ρ nonlin = 0 . 5 ), and a sparse causal structure characterized by a single short path ( κ = 1 , ℓ = 3 ). The causal pathway consists of a binary treatment variable V 71 , a continuous mediator V 35 , a binary mediator V 28 , and a binary outcome V 83 . Despite the limited path depth, the high-dimensional background introduces substantial noise and potential confounding, making this configuration well-suited for testing robustness in pathway recovery under low-signal regimes. It is designed to assess performance when precise signal extraction is required in the presence of many irrelevant or weakly related variables.

V71→V35

V71→V28

V35→V83

V28→V83

1.5

1.0

V71=0.0

1.0

1.0

V28=0.0

1.0

0.8

V71=1.0

0.73

0.8

0.8

V28=1.0

V35

0.5

P(V28|V71)

0.6

0.56

V83

0.6

P(V83V28)

0.6

0.54

0.66

0.0

0.44

0.46

0.4

0.4

0.4

0.34

0.5

0.27

0.2

0.2

0-

Probability(V83=1)

0

1

0.0

0.0

V28

1.0

00

1.0

0.5

0.0

V35

0.5

1.0

1.5

0.0

0.0

V83

1.0

V35→V83|V71

V28→V83|V71

V71→V83

1.0

1.0

V71=0.0

1.0

V71=0.0

V71=1.0

V71=1.0

MediationPathway

0.8

0.8

0.8

V83

0.6

P(V83=1)

0.6

0.61

0.47

0.70

0.59

P(V83}V71)

0.6

0.47

0.66

0.53

Treatment:V71;

0.4

0.4

0.4

0.34

Mediator:V28,V35;

02

Probability(V83=1/V71=0.0)

0.2

0.2

Outcome:V83;

Probability(V83=1(V71=1.0)

00

1.0

-0.5

0.0

V35

0.5

1.0

1.5

0'0

0°0

V28

1.0

0°0

0.0

V83

1.0

00

V71→V176

V71→V35

V71→V57

1.5

1.0

V5=0.0

1.0

V57=0.0

1.0

V5=1.0

0.8

V5=2.0

0.8

V57=2.0

V176

V35

0.0

0.5

P(V5| V71)

06

V5=3.0

P(V57| V71)

0.6

0.5

0.4

0.26 025 0.24 0.25

0.25 026 0.25 0.24

0.4

0.34

0.32

0.34

0.34

0.33

0.33

0.2

0.2

8

1.0

0

0.0

V71=0.0

V71=1.0

0.0

V71=0.0

V71=1.0

V71→V28

V176→V83

V35 →V83

V5→V83

V71=0.0

1.0

1.0

1.0

0.94

0.99

0.8

V71=1.0

0.72

80

0.8

80

08/0

P(V28|V71)

0.62

V83

0.6

V83

0.6

0.48

0.4

0.38

0.4

0.4

0.28

0.2

0.2

Probability(V83=1)

Probability(V83=1)

0.0

0.0

0.0

V28

1.0

-2

V176

。

1.0

0.5

0.0

V35

0.5

1.0

0.0

0.0

1.0

V5

20

3.0

V57→V83

V28→V83

V176 → V83 | V71

V35 → V83 | V71

1.0

1.0

V28=0.0

1.0

1.0

0.8

0.80

0.81

0.78

0.8

V28=1.0

0.80

0.79

0.8

0.8

0.6

0.6

V83

V83

0.4

0.4

0.2

0.20

0.21

0.2

Probability(V83=1|V71=0.0)

Probability(V83=1{V71=0.0)

Probability(V83=1(V71=1.0)

Probability(V83=1/V71=1.0)

0.0

0.0

0.0

0.0

V57

1.0

20

0.0

0.0

V83

1.0

-2

V176

1.0

-0.5

V35

0.5

1.0

V57 →V83|V71

V28 →V83|V71

V71→ V83

V71=0.0

0.95

0.92

0.99 0.98

1.0

V71=0.0

1.0

V71=0.0

1.0

V71=0.0

08

V71=1.0

0.74

0.8

V71=1.0 0.83

0.79

0.79

0.76

0.8

V71=1.0

0.81

0.77

0.8

V71=1.0

0.82

0.77

P(V83=1)

0.6

P(V83}V71)

0.41

0.4

0.2

0.2

0.2

0.2

0.18

0.23

0.0

0.0

1.0

V5

2.0

3.0

0.0

0.0

V57

1.0

2.0

0.0

0.0

V28

1.0

0.0

0.0

V83

1.0

02

V35→V72

V35→V2

V35→V68

V35→V14

V2=0.0

1.0

V2=1.0

0.5

V2=2.0

0.0

P(V2| V35)

V2=3.0

0.5

12

90

V68

V14

0l-

-1.5

0.25 0.25 0.24 0.26

0.25 0.25 0.25 0.25

2.0

0.2

-2.5

3.0

0

V35

V35=0.0

V35

V35=1.0

V35

V35

V72 →V17

V2→V17

V68→V17

V14 → V17

V17

V17

V17

E-

-2

V72

-1

1

2

V2

-1

V68

0

3.0

-2.5-2.0

-1.5-1.0

V14

0.5

0.0

0.5

V72→V17|V35

V2→V17|V35

V68→V17|V35

V14 → V17| V35

■

V35=0.0

■

V35=0.0

V35=0.0

■

V35=0.0

V35=1.0

V35=1.0

V35=1.0

V35=1.0

~4

3

-2

V72

2

V2=0.0

V21.0

V2

V2=2.0

V2=3.0

V68

-3.0-2.5-2.0-1.5-1.0

V14

-0.50.00.51.0

V35→V17

MediationPathway

Treatment:V35;

Mediator: V2,V14,V68,V72;

Outcome: V17;

V35

V35→V48

V35→V33

V35 →V6

V35→V42

5.0

7.5

5.0

2.5

5.0

2.5

5

00

25

0.0

-2.5-

0.0

V42

15

5.0

2.5

-5.0

20

8

7.5

5.0

-7.5

25

1.0

12.5

8

7.5

10.0

12.5

0

V35

。

V35

0

V35

0

V35

V35→V14

V48 → V45

V33→V45

V6 → V45

1.

10

1.0

5.0

000b

2.5

0.8

80

0.8

0.0

0.6

0.6

0.6

2.5

V45

5.0

.4

0.4

0.A

7.5

0.2

02

02

。

10.0

8

。

0.0

Probability(V45=1)

00

Probability(V45=1)

0.0

Probability(V45=1)

V35

1

-2520

-15

-10

V48

-5

=12.5

10.0

-7.5

5.0

2.5

V33

0.0

2.5

5.0

-7.5

-5.0

2.5

0.0

V6

25

5.0

7.5

V14 →V45

V48 →V45 |V35

V33 →V45|V35

1.0

1.0

1.0

0.8

0.8

0.8

0.8

V45

0.6

0.6

V45

0.6

0.

0.4

0.4

0.4

02

.2

0.2

Probability(V45=1/V35=0.0)

Probability(V45=1IV35=0.0)

Probability(V45=1)

Probability(V45=1)

Probability(V45=1/V35=1.0)

Probability(V45=1/V35=1.0)

0.0

0.0

0.0

0.0

-12.5 10.0-7.5

5.0

-25

0.0

255.0

-10.07.5

-2.5

0.0

2.5

5.0

-25

20

-15

-10

-5

。5

12.5

10.0-7.5

-5.0-2.5

0.0255.0

V42

V14

V48

V33

V6→V45| V35

V42 →V45| V35

V14 →V45|V35

V35→V45

1.0

1.0

10

1.0

V35=0.0

V35=1.0

0.8

.8

0.8

0.8

0.68

V45

0.6

V45

0.6

V45

0.6

0.58

0.4

0.4

0.42

0.32

Probability(V45=1|V35=0.0)

0.2

Probability(V45=1/V35=0.0)

0.2

Probability(V45=1/V35=0.0)

0.2

Probability(V45=1/V35=1.0)

Probability(V45=1/V35=1.0)

Probability(V45=1/V35=1.0)

00

7.5

-5.0

-2.5

0.0

V6

2.5

5.0

7.5

0.0

12.5

-10.0

-7.5

5.0

V42

-2.5

00

2.5

0.0

10.0

-7.5

5.0

V14

-2.5

0.0

2.5

5.0

0.0

0.0

V45

F Experiment

F.1 Experiment Setup

In the structure discovery phase, we configure the Flow-SEM model with a two-layer MLP (hid-den_dim_multiplier=2) as the conditional distribution model. The model is trained using the Adam optimizer with a learning rate of 0.001, an initial DAG constraint penalty coefficient λ = 0 . 1 , which increases by a factor of 10 when the acyclicity constraint is violated (h_threshold= 10 -8 ), up to a maximum of 10 5 . Sparsity is controlled through L1 regularization ( α = 0 . 01 ), with a maximum gradient norm limit of 1.0. The training iterates for 1000 epochs, with early stopping if no improvement occurs for 100 consecutive epochs.

In the posterior sampling phase, we extract 1000 DAG samples from the learned structure. During sampling, each node retains only the top 3 highest probability incoming edges to control sparseness, and a sigmoid function converts the weight matrix to edge existence probabilities. Graph structures are randomly sampled according to a Bernoulli distribution, validated for acyclicity through topological sorting, and cyclic graphs are discarded.

For path identification, we set the frequency threshold to 5%, meaning paths appearing with frequency above this threshold in the DAG sample set are identified as stable paths. The maximum path length is limited to 10, and multi-processor parallel computing is employed to accelerate the identification process.

The mechanism modeling phase implements 5-fold cross-validation. Each model uses uniform parameters: 50 iterations, batch size of 128, learning rate of 10 -4 , and hidden layer dimension of 128. For continuous treatment variables, intervention levels are set at the mean ( µ ) and the mean plus one standard deviation ( µ + σ ), while binary variables use intervention levels {0,1}.

Effect estimation employs a mixed approach: paths of length 3 use Efficient Influence Function estimation, while longer paths use plugin estimation. Multiple DAG effects are aggregated through Bayesian Model Averaging, considering both within-DAG estimation uncertainty (average variance) and between-DAG structural uncertainty (effect variation), constructing 95% confidence intervals.

We test SIGMA on 10 synthetic datasets with feature dimensions p ∈ { 20 , 50 , 100 , 200 } , nonlinearity degree ρ ∈ { 0 , 0 . 5 } , containing parallel ( κ ∈ { 2 , 3 , 6 } ) and chain ( ℓ ∈ { 1 , 2 } ) structures.

F.2 Computation of Ground Truth Mediation Effects

To validate SIGMA's estimation accuracy, we compute the true NDE and NIE values for the two mediation pathways through counterfactual simulation. For a mediation pathway X → M → Y , we define three counterfactual data sets: D ref = { ( X i = x ref , M i ( x ref ) , Y i ( x ref , M i ( x ref ))) } n i =1 , D comp = { ( X i = x comp , M i ( x comp ) , Y i ( x comp , M i ( x comp ))) } n i =1 , and D cf = { ( X i = x comp , M i ( x ref ) , Y i ( x comp , M i ( x ref ))) } n i =1 , where M i ( x ) denotes the potential mediator value when treatment X i = x , and Y i ( x, m ) represents the potential outcome when X i = x and M i = m .

Based on these counterfactual sets, the true mediation effects are calculated as: NDE = E [ Y ( x comp , M ( x ref ))] -E [ Y ( x ref , M ( x ref ))] and NIE = E [ Y ( x comp , M ( x comp ))] -E [ Y ( x comp , M ( x ref ))] . In Monte Carlo simulation, these expectations are estimated via sample means: ̂ NDE = 1 n ∑ n i =1 Y i ( x comp , M i ( x ref )) -1 n ∑ n i =1 Y i ( x ref , M i ( x ref )) and ̂ NIE = 1 n ∑ n i =1 Y i ( x comp , M i ( x comp )) -1 n ∑ n i =1 Y i ( x comp , M i ( x ref )) .

Wecompute these effects using structural equation model specifications with 5000 simulation samples, setting intervention levels for pathway 1 (V5 → V19 → V16) at x ref = 0 . 012 and x comp = 1 . 253 , and for pathway 2 (V12 → V10 → V6) at x ref = 0 . 243 and x comp = 0 . 926 . To generate D cf , we extract M i ( x ref ) values from D ref , then set X i = x comp and fix M i = M i ( x ref ) , employing topological sorting to ensure proper variable generation order in the DAG.

F.3 Causal Discovery Evaluation

F.4 Real Data HPP Validation

ForestPlotofMediationEffects

2 -->1 --> 6

3 --> 1 --> 12 --> 13

3 --> 1 --> 12

4 --> 12 --> 6

2 --> 1 --> 12 --> 6

3 -->12 --> 6

4 --> 1 --> 12 --> 13

4 --> 12 --> 13

2 --> 1 --> 12 --> 13

4 --> 1 --> 12

4 --> 1 --> 6

3 --> 1 -->6

Total Effect

Direct Effect

2 --> 1 --> 12

Indirect Effect

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

EffectSize

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes an end-to-end framework called SIGMA (Structure-Informed Guided Mediation Analysis), which aims to address the disconnection between structure discovery and the understanding of mediation mechanisms in existing causal mediation analysis. SIGMA integrates three key steps: probabilistic causal structure discovery, automated mediation pathway identification, and effect estimation, while systematically quantifying and propagating structural uncertainty. Specifically, the framework first utilizes differentiable Flow-Structural Equation Models (Flow-SEM) to learn the structural posterior distribution, generating diverse Directed Acyclic Graphs (DAGs) to quantify structural uncertainty. Based on these DAGs, the paper introduces the Path Stability Score (PSS) to evaluate the marginal probability of pathways, thereby identifying high-confidence mediation paths. For the identified pathways, SIGMA combines Efficient Influence Functions (EIF) and Bayesian model averaging (BMA) to estimate pathway effects, propagating both within-structure estimation uncertainty and between-structure effect variation to the final effect estimates.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n1.  This paper proposes a unified framework (SIGMA) that integrates causal structure discovery, mediation pathway identification, and effect estimation, effectively addressing the problems caused by the separation of these stages in traditional methods.\n2.  The SIGMA framework combines multiple advanced statistical and machine learning techniques, such as Flow-SEM, EIF, BMA, and a Variational Autoencoder (VAE) imputation module, providing a relatively solid theoretical foundation. The authors have also provided quite detailed proofs for the core theoretical basis of this paper.\n3.  This paper conducts experiments on both real-world and synthetic datasets. The experiments are presented from relatively sufficient perspectives, including linear/non-linear scenarios and different data scales, which can effectively demonstrate the feasibility of its method.\n\n**Weaknesses:**\n\n1.  The SIGMA framework may contain multiple computationally intensive steps, making an analysis of its time complexity necessary. The authors should at least provide running times on datasets of different scales to illustrate the relationship between time consumption and data size.\n2.  The synthetic datasets mentioned in the paper are all constructed based on the ER model; however, real-world data often do not entirely conform to the distribution of ER models. The authors should attempt experiments on different models (e.g., SF).\n\nQUESTIONS:\n1.  As mentioned in the weaknesses, some experiments could be supplemented.\n2.  In Appendix F1, it is stated that the frequency threshold is set to 5%. Is this \"frequency threshold\" the same as the τ mentioned earlier in the text? If so, why was τ valued at 0.15 previously and 0.05 here?\n3.  I have difficulty understanding the descriptions related to Section 5.3 and Figure 3. For example, I did not see the path \"sleep HRV index → pulse wave velocity\" 【line:324】described in the text explicitly labeled in Figure 3. Could the description be made clearer?\n4.  \"structures uncertainty\" 【line:22】should be \"structural uncertainty\".\n5. The author is requested to carefully check and confirm the usage of specialized terms within the manuscript. For example, a given specialized term should be written in its full form only at its first mention (as is the case with BMA and PSS in the text). Additionally, the capitalization (of initial letters) for these terms should be consistent (e.g., Path Stability Score, NOTEARS), and so on.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the challenge of handling structural uncertainty in Causal Mediation Analysis (CMA). Traditional CMA methods either assume the causal structure is known, making them unable to handle unknown structures, or they fail to provide consistent inference that properly accounts for structural uncertainty, even when the structure is estimated. To overcome these limitations, the authors propose a novel framework called SIGMA (Structure-Informed Guided Mediation Analysis). SIGMA leverages a Flow-Structural Equation Model (Flow-SEM) to model the posterior probabilities of edges in possible Directed Acyclic Graphs (DAGs). It introduces a new metric, Path Stability Score (PSS), to identify high-confidence mediation pathways. Effect estimation is then performed based on these stable pathways.\n\nExperiments on synthetic datasets demonstrate that SIGMA achieves more accurate structure recovery than conventional causal discovery methods and more reliable mediation path identification compared to existing CMA approaches. Furthermore, application to real-world data related to sleep and cardiovascular health shows that SIGMA can successfully reconstruct known mediation mechanisms and uncover previously unreported pathways, highlighting its potential for new scientific discovery.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe paper presents a highly robust and reliable framework for Causal Mediation Analysis (CMA) by incorporating structural uncertainty into effect estimation through the combination of Flow-SEM and Path Stability Score (PSS), effectively leveraging a relaxed acyclicity constraint. This integration enables a novel end-to-end treatment of uncertainty that surpasses prior approaches. The effectiveness of the proposed method is quantitatively demonstrated through comprehensive experiments on synthetic data. Furthermore, its applicability and potential for scientific discovery are substantiated via real-world evaluation using sleep-related physiological data.\n\nWeaknesses:\nWhile the proposed treatment of structural uncertainty is conceptually compelling and methodologically sound, the paper does not include a comparison against baseline methods that incorporate structural uncertainty via bootstrap-style approaches, which are feasible even for conventional causal discovery algorithms. Although it is intuitively and theoretically plausible that modeling structural uncertainty directly within the learning framework, as done in this paper, yields superior results compared to such post-hoc strategies, empirical validation of this advantage would substantially strengthen the paper's claims. Including such a comparison in the synthetic experiments would significantly enhance the persuasiveness and completeness of the evaluation.\n\nQUESTIONS:\n1. Would it be possible to compare the proposed method against conventional structure learning methods that are enhanced with bootstrap-style techniques to account for structural uncertainty? If the proposed approach can be shown to outperform such alternatives, it would significantly strengthen the overall evaluation of the paper.\n\n2. For the novel mediation pathway shown in Figure 3(c), is it possible to cite any prior studies, however minimal, that suggest the plausibility of the entire pathway as a coherent mechanism? While the paper acknowledges that individual links have been previously studied, it also states that the complete pathway has not been reported as such. If even partial evidence supporting the full path exists, the method could be seen as successfully identifying an under-recognized yet plausible hypothesis. This would highlight the proposed approach as a powerful tool for supporting scientific discovery and could substantially elevate the contribution of the paper.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a new method for mediation analysis called SIGMA, which has three main steps: causal structure learning, mediation path extraction, and estimation of the natural direct and indirect effects. SIGMA is end-to-end, and the estimated effects account for uncertainty about the causal structure. SIGMA is compared to other algorithms on synthetic data and also used to analyze a real health dataset.\n\nSTRENGTHS AND WEAKNESSES:\n*Quality*:\n- (+) the paper is comprehensive, touching upon many ideas from causal discovery and inference as well as its health application\n- (-) however, technical depth and rigor is lacking, with some important concepts left ambigous, undefined, or conflated (see Question 1)\n- (-) state-of-the-art causal discovery methods (like BOSS and order MCMC; see [here](https://benchpressdocs.readthedocs.io/en/latest/examples.html#study-4-fpr-tpr-pattern-png) for more details) aren't used in comparisons; for the effect estimation task, there's no comparison to an obvious (causal discovery-based) baseline like using the MLE for the output of LiNGAM\n\n*Clarity*: (+) the paper is generally well-written ([-] with the important exception of some technical imprecision)\n\n*Significance*: (+) there's a lot of potential value in an end-to-end mediation analysis that accounts for structural uncertainty, but (-) I don't see much theoretical significance here, and (-) the practical significance remains unclear based on the current experiments\n\n*Originality*: (+) the approach for explicitly accounting for structural uncertainty in the causal effect estimate is novel (and seems to sensibly builds off/connects to some existing work in an interesting way); but (-) it's not always so clear exactly what is novel versus what is existing work, e.g. Section 4 contains no references at all\n\nQUESTIONS:\n1. What does \"identify\" (and all the variations on this root) mean in this paper? Within the causal inference and discovery literature, I'm used to seeing \"identify\" in the formal sense of expressing a causal estimand in terms of observed distributions. This paper seems to mostly (but not always) mean something more like \"estimate\". While identifiability is required for unbiased estimation, these are distinct concepts, and conflating them detracts from the rigor and clarity of the paper. (A similar point can be made for \"bias\" versus \"error\", e.g., on line 237 and in Table 1.)\n\nAs a follow-up, is the paper presenting/making use of new identifiability results for mediation paths beyond those implied by the usual causal discovery setup? I don't see an explicit formal statement of this anywhere, but it seems to be implied (e.g. top of Page 2).\n\n2. Can the authors give some sense of the complexity or runtime of the proposed method? This is important to understand how practical it actually is. Ideally, there would be explicit statements/proofs of the complexity along with runtime comparisons to other algorithms.\n\n3. How do the SIGMA results on the real dataset compare to other methods? The current results suggest SIGMA is reasonable but they don't demonstrate its significance compared to existing methods.\n\n*Minor comments*:\n1. L. 111: should cite the original Erdős and Rényi paper rather than an arbitrary recent causal discovery paper.\n2. Section 3: would be helpful to relate these to the Markov and faithfulness assumptions---for example, isn't conditional ignorability satisfied when conditioning on the parents of A as described in Section 4.4.1?\n3. L. 158: cofounders -> covariates?\n4. L. 322--323: \"successfully identified multiple physiologically meaningful mediation pathways\" seems a bit strong since there's no ground truth; something along the lines of \"learned multiple physiologically plausible pathways\" seems to be all that's justfied based on the current results\n5. I don't really get the distinction between DECI and the others in lines 349--351. For example, GES uses the MLE (via the BIC score) for structure learning, so isn't it also jointly inferring causal structure and structural equations?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents SIGMA, a unified framework for causal mediation analysis. The authors identify that traditional causal mediation suffers from the issue of trying to balance *principled mechanisms for quantifying structural uncertainty* (interpretability/prior-informed) with *efficient Pathway Identification* (accuracy/performance). They build SIGMA by integrating three key components/functionalities together: structure discovery, pathway identification, and effect estimation. \n\nSIGMA uses flow-based structural equation models to learn uncertain causal structures, samples an ensemble of DAGs, and then introduces a Path Stability Score to identify high-confidence mediation pathways across those DAGs—avoiding the need for manual path specification. For each pathway, the authors also provide a Bayesian sampling inspired approach for quantifying structural uncertainty in mediation effects. The authors provide an interesting simulation study as well as a real-world application to HPP data, revealing novel mediation pathways linking sleep quality to cardiovascular health.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n- The authors present a strong theoretical foundation (sections 3 and 4) for the pathway, mediation, and DAG estimation\n- Interesting design of simulation experiments\n- Meaningful comparisons to competing methodologies\n\nWeaknesses (elaborated below): \n-  The additive noise SEMs in simulation studies have a simplistic form (Appendix E1) which may be unrealistic \n-  The robustness of the method to noise is not clear\n-  The scalability of the model is questionable\n-  Need more real world evidence\n\nQUESTIONS:\nI have four main concerns, which - if addressed - can substantially improve the paper, especially from an applied perspective: \n\n**Simplistic SEMs in Simulation Studies:**\nThe additive noise structural equation models (SEMs) used in the simulations (Appendix E1) appear to have relatively simplistic functional forms. This raises the concern that SIGMA's strong performance may be driven in part by the particular choice of SEMs, rather than its general robustness. It would be helpful if the authors could test SIGMA under more complex or nonlinear SEMs, or at least provide a stronger justification for why the current SEM choices reflect realistic data-generating processes.\n\n**Unclear Robustness to Noise:**\nMethods of this kind often degrade with increasing observational noise, yet it's not clear how sensitive SIGMA is to increasing noise levels. Including such a robustness analysis would help clarify how noise-tolerant SIGMA is, and where its limitations lie. \n\n**Scalability to High-Dimensional Settings:**\nWhile the simulation studies do explore dimensionalities up to 200 variables, many real-world systems—such as gene regulatory networks—involve thousands of variables. It's unclear whether SIGMA remains computationally feasible or accurate at such scales. I encourage the authors to evaluate its performance on higher-dimensional data (e.g., 500, 1000, or 5000 variables), as this would greatly help readers assess its applicability across domains.\n\n**Real-World Application:**\nFinally, while I appreciate the depth of the theoretical development, I think the paper would benefit from a second, more large-scale real-world application. For instance, applying SIGMA to a large-scale gene regulatory dataset could be a strong demonstration of its practical utility. In particular, evaluating whether SIGMA can recover known mediation pathways in well-studied biological circuits would offer a more tangible result for potential users in biology.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}