Structural Causal Models

Also known as SCM

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Overview

Structural causal models represent how variables are generated by mechanisms, often organized in a graph, so they can answer questions about interventions and counterfactuals rather than correlations alone. Researchers learn these models from observational and experimental data using constraint- and score-based graph discovery, linear and non-Gaussian structural equation models, nonlinear neural or symbolic mechanisms, and invariance across environments; latent confounding, feedback, and time-varying systems extend the basic acyclic setting. The models support treatment-effect estimation, scientific mechanism analysis, fairness and recourse, anomaly diagnosis, and causal world models, with evaluation increasingly using synthetic benchmarks and intervention-based tests. A central challenge is that observational data often leave multiple structures or counterfactuals compatible with the evidence, so progress depends on explicit assumptions, informative interventions, and reliable uncertainty and benchmark evaluation.

Latest papers

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144 papers16 papers in the last four weeks, up 7% on the four weeks before. 0.1% of all new papers.

Apr 5, 2026

DAGAF: A directed acyclic generative adversarial framework for joint structure learning and tabular data synthesis

Understanding the causal relationships between data variables can provide crucial insights into the construction of tabular datasets. Most existing causality learning methods typically focus on applying a single identifiable causal model, such as the Additive Noise Model (ANM) or the Linear non-Gaussian Acyclic Model (LiNGAM), to discover the dependencies exhibited in observational data. We improve on this approach by introducing a novel dual-step framework capable of performing both causal structure learning and tabular data synthesis under multiple causal model assumptions. Our approach uses Directed Acyclic Graphs (DAG) to represent causal relationships among data variables. By applying various functional causal models including ANM, LiNGAM and the Post-Nonlinear model (PNL), we implicitly learn the contents of DAG to simulate the generative process of observational data, effectively replicating the real data distribution. This is supported by a theoretical analysis to explain the multiple loss terms comprising the objective function of the framework. Experimental results demonstrate that DAGAF outperforms many existing methods in structure learning, achieving significantly lower Structural Hamming Distance (SHD) scores across both real-world and benchmark datasets (Sachs: 47%, Child: 11%, Hailfinder: 5%, Pathfinder: 7% improvement compared to state-of-the-art), while being able to produce diverse, high-quality samples.

Hristo Petkov, Calum MacLellan, Feng Dong

Department of Computer and Information Sciences, University of Strathclyde, 16 Richmond Street, Glasgow, G1 1XQ, Lanarkshire, United Kingdom.

Causal Structure LearningTabular Data SynthesisStructural Causal ModelsGenerative Adversarial Networks
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Apr 1, 2026

Procela: Epistemic Governance in Mechanistic Simulations Under Structural Uncertainty

Mechanistic simulations typically assume fixed ontologies: variables, causal relationships, and resolution policies are static. This assumption fails when the true causal structure is contested or unidentifiable-as in antimicrobial resistance (AMR) spread, where contact, environmental, and selection ontologies compete. We introduce Procela, a Python framework where variables act as epistemic authorities that maintain complete hypothesis memory, mechanisms encode competing ontologies as causal units, and governance observes epistemic signals and mutates system topology at runtime. This is the first framework where simulations test their own assumptions. We instantiate Procela for AMR in a hospital network with three competing families. Governance detects coverage decay, policy fragility, and runs structural probes. Results show 20.4% error reduction and 69% cumulative regret improvement over baseline. All experiments are reproducible with full auditability. Procela establishes a new paradigm: simulations that model not only the world but their own modeling process, enabling adaptation under structural uncertainty.

Kinson Vernet

Independent Researcher

Epistemic UncertaintyUncertainty QuantificationStructural Causal Models
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Mar 28, 2026

On the Relationship between Bayesian Networks and Probabilistic Structural Causal Models

In this paper, the relationship between probabilistic graphical models, in particular Bayesian networks, and causal diagrams, also called structural causal models, is studied. Structural causal models are deterministic models, based on structural equations or functions, that can be provided with uncertainty by adding independent, unobserved random variables to the models, equipped with probability distributions. One question that arises is whether a Bayesian network that has obtained from expert knowledge or learnt from data can be mapped to a probabilistic structural causal model, and whether or not this has consequences for the network structure and probability distribution. We show that linear algebra and linear programming offer key methods for the transformation, and examine properties for the existence and uniqueness of solutions based on dimensions of the probabilistic structural model. Finally, we examine in what way the semantics of the models is affected by this transformation. Keywords: Causality, probabilistic structural causal models, Bayesian networks, linear algebra, experimental software.

Peter J. F. Lucas, Eleonora Zullo, Fabio Stella

ICIS, Radboud University, Nijmegen, the Netherlands · Dipartimento di Informatica, Sistemistica e comunicazion University of Milano-Bicocca, Milan, Italy

Structural Causal ModelsBayesian NetworksProbabilistic Graphical Models
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Mar 20, 2026

Kolmogorov-Arnold causal generative models

Causal generative models provide a principled framework for answering observational, interventional, and counterfactual queries from observational data. However, many deep causal models rely on highly expressive architectures with opaque mechanisms, limiting auditability in high-stakes domains. We propose KaCGM, a causal generative model for mixed-type tabular data where each structural equation is parameterized by a Kolmogorov--Arnold Network (KAN). This decomposition enables direct inspection of learned causal mechanisms, including symbolic approximations and visualization of parent--child relationships, while preserving query-agnostic generative semantics. We introduce a validation pipeline based on distributional matching and independence diagnostics of inferred exogenous variables, allowing assessment using observational data alone. Experiments on synthetic and semi-synthetic benchmarks show competitive performance against state-of-the-art methods. A real-world cardiovascular case study further demonstrates the extraction of simplified structural equations and interpretable causal effects. These results suggest that expressive causal generative modeling and functional transparency can be achieved jointly, supporting trustworthy deployment in tabular decision-making settings. Code: https://github.com/aalmodovares/kacgm

Alejandro Almodóvar, Mar Elizo, Patricia A. Apellániz +2

Generative ModelingStructural Causal ModelsCausal Interpretability
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Mar 19, 2026

Teleological Inference in Structural Causal Models via Intentional Interventions

Structural causal models (SCMs) were conceived to formulate and answer causal questions. This paper shows that SCMs can also be used to formulate and answer teleological questions, concerning the intentions of a state-aware, goal-directed agent intervening in a causal system. We review limitations of previous approaches to modeling such agents, and then introduce intentional interventions, a new time-agnostic operator that induces a twin SCM we call a structural final model (SFM). SFMs treat observed values as the outcome of intentional interventions and relate them to the counterfactual conditions of those interventions (what would have happened had the agent not intervened). We show how SFMs can be used to empirically detect agents and to discover their intentions.

Dario Compagno, Fabio Massimo Zennaro

Structural Causal ModelsCausal Intervention
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Mar 18, 2026

Cohomological Obstructions to Global Counterfactuals: A Sheaf-Theoretic Foundation for Generative Causal Models

Current continuous generative models (e.g., Diffusion Models, Flow Matching) implicitly assume that locally consistent causal mechanisms naturally yield globally coherent counterfactuals. In this paper, we prove that this assumption fails fundamentally when the causal graph exhibits non-trivial homology (e.g., structural conflicts or hidden confounders). We formalize structural causal models as cellular sheaves over Wasserstein spaces, providing a strict algebraic topological definition of cohomological obstructions in measure spaces. To ensure computational tractability and avoid deterministic singularities (which we define as manifold tearing), we introduce entropic regularization and derive the Entropic Wasserstein Causal Sheaf Laplacian, a novel system of coupled non-linear Fokker-Planck equations. Crucially, we prove an entropic pullback lemma for the first variation of pushforward measures. By integrating this with the Implicit Function Theorem (IFT) on Sinkhorn optimality conditions, we establish a direct algorithmic bridge to automatic differentiation (VJP), achieving O(1)-memory reverse-mode gradients strictly independent of the iteration horizon. Empirically, our framework successfully leverages thermodynamic noise to navigate topological barriers ("entropic tunneling") in high-dimensional scRNA-seq counterfactuals. Finally, we invert this theoretical framework to introduce the Topological Causal Score, demonstrating that our Sheaf Laplacian acts as a highly sensitive algebraic detector for topology-aware causal discovery.

Rui Wu, Hong Xie, Yongjun Li

School of Management, University of Science and Technology of China · School of Computer Science and Engineering, University of Science and Technology of China

Causal Counterfactual GenerationCausal InferenceStructural Causal ModelsCausal Discovery
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Mar 12, 2026

Multi-Task Anti-Causal Learning for Reconstructing Urban Events from Residents' Reports

Many real-world machine learning tasks are anti-causal: they require inferring latent causes from observed effects. In practice, we often face multiple related tasks where the structural dependencies are a hybrid of task-invariant and task-specific mechanisms. We propose Multi-Task Anti-Causal learning (MTAC), a framework for estimating causes from outcomes and confounders by explicitly exploiting such cross-task invariances. MTAC learns a structural equation model (SEM) that factorizes the outcome-generation process into (i) a task-invariant mechanism and (ii) task-specific mechanisms via a shared backbone with task-specific deviations. Building on the learned forward model, MTAC performs maximum A posteriori (MAP) based inference to reconstruct causes by jointly optimizing latent mechanism variables and cause magnitudes under the learned structural model. We evaluate MTAC on the application of urban event reconstruction from resident reports, spanning three tasks: parking violations, abandoned properties, and unsanitary conditions. On real-world data collected from Manhattan and Newark, MTAC consistently improves reconstruction accuracy over strong baselines, achieving up to 33.04% MAE reduction and demonstrating the benefits of learning transferable mechanisms across tasks.

Liangkai Zhou, Susu Xu, Shuqi Zhong +1

Stony Brook University Stony Brook, New York, United States · Johns Hopkins University Baltimore, Maryland, United States

Structural Causal ModelsMulti-Task LearningCausal Inference
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Mar 11, 2026

Interventional Time Series Priors for Causal Foundation Models

Prior-data fitted networks (PFNs) have emerged as powerful foundation models for tabular causal inference, yet their extension to time series remains limited by the absence of synthetic data generators that provide interventional targets. Existing time series benchmarks generate observational data with ground-truth causal graphs but lack the interventional data required for training causal foundation models. To address this, we propose CausalTimePrior, a principled framework for generating synthetic temporal structural causal models (TSCMs) with paired observational and interventional time series. Our prior supports configurable causal graph structures, nonlinear autoregressive mechanisms, regime-switching dynamics, and multiple intervention types (hard, soft, time-varying). We demonstrate that PFNs trained on CausalTimePrior can perform in-context causal effect estimation on held-out TSCMs, establishing a pathway toward foundation models for time series causal inference.

Dennis Thumm, Ying Chen

National University of Singapore · Department of Mathematics, Centre for Quantitative Finance, Risk Management Institute, National University of Singapore

Causal Foundation ModelsSynthetic Data GenerationCausal Effect EstimationStructural Causal Models
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Mar 9, 2026

Structural Causal Bottleneck Models

We introduce structural causal bottleneck models (SCBMs), a novel class of structural causal models in which causal effects between high-dimensional variables are mediated by low-dimensional summary statistics, or bottlenecks. SCBMs provide a flexible framework for mechanism-specific, target-dependent dimension reduction while remaining estimable via standard learning algorithms. We prove that bottleneck variables are identifiable up to bijection from observational data, and validate this experimentally. We then study causal effect estimation in linear SCBMs under three standard adjustment strategies. For instrumental variables, we identify a population-level non-identifiability of naïve two-stage least squares when all variables are high-dimensional, and show that bottleneck representations restore identification. For mediation and backdoor adjustment, we prove that conditioning on bottleneck variables yields valid effect estimates with explicit finite-sample efficiency gains over conditioning on full high-dimensional variables. We argue that SCBMs provide a principled alternative to existing causal dimension reduction frameworks such as causal representation learning and causal abstractions.

Simon Bing, Jonas Wahl, Jakob Runge

Equal contribution · Technische Universität Berlin · Department of Computer Science, University of Potsdam +1

Structural Causal ModelsCausal Effect EstimationCausal Identifiability
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Mar 9, 2026

Sign Identifiability of Causal Effects in Stationary Stochastic Dynamical Systems

We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.

Gijs van Seeventer, Saber Salehkaleybar

1LIACS, Leiden University, Netherlands

Stochastic Differential EquationsStructural Causal ModelsCausal Identifiability
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Mar 5, 2026

Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning

Causal discovery with latent variables is a fundamental task. Yet most existing methods rely on strong structural assumptions, such as enforcing specific indicator patterns for latents or restricting how they can interact with others. We argue that a core obstacle to a general, structural-assumption-free approach is the lack of an equivalence characterization: without knowing what can be identified, one generally cannot design methods for how to identify it. In this work, we aim to close this gap for linear non-Gaussian models. We establish the graphical criterion for when two graphs with arbitrary latent structure and cycles are distributionally equivalent, that is, they induce the same observed distribution set. Key to our approach is a new tool, edge rank constraints, which fills a missing piece in the toolbox for latent-variable causal discovery in even broader settings. We further provide a procedure to traverse the whole equivalence class and develop an algorithm to recover models from data up to such equivalence. To our knowledge, this is the first equivalence characterization with latent variables in any parametric setting without structural assumptions, and hence the first structural-assumption-free discovery method. Code and an interactive demo are available at https://equiv.cc.

Haoyue Dai, Immanuel Albrecht, Peter Spirtes +1

Carnegie Mellon University · FernUniversität in Hagen · MBZUAI

Causal DiscoveryCausal Discovery with Latent ConfoundingCausal IdentifiabilityStructural Causal Models
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Mar 3, 2026

Scalable Contrastive Causal Discovery under Unknown Soft Interventions

Observational causal discovery is only identifiable up to the Markov equivalence class. While interventions can reduce this ambiguity, in practice interventions are often soft with multiple unknown targets. In many realistic scenarios, only a single intervention regime is observed. We propose a scalable causal discovery model for paired observational and interventional settings with shared underlying causal structure and unknown soft interventions. The model aggregates subset-level PDAGs and applies contrastive cross-regime orientation rules to construct a globally consistent maximal PDAG under Meek closure, enabling generalization to both in-distribution and out-of-distribution settings. Theoretically, we prove that our model is sound with respect to a restricted ΨΨ equivalence class induced solely by the information available in the subset-restricted setting. We further show that the model asymptotically recovers the corresponding identifiable PDAG and can orient additional edges compared to non-contrastive subset-restricted methods. Experiments on synthetic data demonstrate improved causal structure recovery, generalization to unseen graphs with held-out causal mechanisms, and scalability to larger graphs, with ablations supporting the theoretical results.

Mingxuan Zhang, Khushi Desai, Sopho Kevlishvili +1

Department of Systems Biology, Columbia University · Irving Institute for Cancer Dynamics, Columbia University · Department of Computer Science, Columbia University +2

Causal DiscoveryStructural Causal ModelsCausal IdentifiabilityCausal Intervention
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Mar 1, 2026

SphUnc: Hyperspherical Uncertainty Decomposition and Causal Identification via Information Geometry

Reliable decision-making in complex multi-agent systems requires calibrated predictions and interpretable uncertainty. We introduce SphUnc, a unified framework combining hyperspherical representation learning with structural causal modeling. The model maps features to unit hypersphere latents using von Mises-Fisher distributions, decomposing uncertainty into epistemic and aleatoric components through information-geometric fusion. A structural causal model on spherical latents enables directed influence identification and interventional reasoning via sample-based simulation. Empirical evaluations on social and affective benchmarks demonstrate improved accuracy, better calibration, and interpretable causal signals, establishing a geometric-causal foundation for uncertainty-aware reasoning in multi-agent settings with higher-order interactions.

Rong Fu, Chunlei Meng, Jinshuo Liu +8

Independent Researcher

Uncertainty DecompositionUncertainty CalibrationMulti-Agent ReasoningCausal IdentifiabilityStructural Causal ModelsHyperspherical Representation Learning
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Feb 27, 2026

Causal Mechanism Reduction: Mechanism Replacement for Neural Network Pruning and Abstraction

Which internal mechanisms of a neural network can be replaced while preserving the computation it performs? Structured pruning asks for smaller deployable networks; causal abstraction asks for high-level models that commute with interventions. We introduce causal mechanism reduction (CMR), a framework that treats a trained network as a deterministic structural causal model and replaces selected internal variables by constants or affine functions of retained variables. These replacements compile exactly into smaller dense networks by bias and weight folding, and induce reduced causal models testable with interchange interventions. We derive a unified second-order replacement-risk objective whose special cases recover mean replacement, variance-based pruning (VBP), logit-distortion scoring, and affine neuron merging, together with a margin-based certificate linking logit distortion to interchange-intervention agreement. The framework also exposes a basic invariance requirement: functionally identical ReLU networks should induce the same reduction. Under exact positive-scaling reparameterizations, VBP's kept set collapses to chance-level overlap while the logit-distortion score is exactly invariant. Empirically, CMR variants are competitive with VBP under matched fine-tuning of DeiT-Tiny on ImageNet-100; the clearer separation appears in the invariance and interchange tests, where the logit-distortion score preserves kept sets and consistently improves distributional fidelity. CMR thus gives pruning, compilation, and causal-abstraction verification a common object to optimize and verify.

Amir Asiaee

Vanderbilt University Medical Center Nashville, TN

Structural Causal ModelsCausal AbstractionNeural Network CompressionNeural Network Pruning
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Feb 27, 2026

Operationalizing Longitudinal Causal Discovery Under Real-World Workflow Constraints

Causal discovery has achieved substantial theoretical progress, yet its deployment in large-scale longitudinal systems remains limited. A key obstacle is that operational data are generated under institutional workflows whose induced partial orders are rarely formalized, enlarging the admissible graph space in ways inconsistent with the recording process. We characterize a workflow-induced constraint class for longitudinal causal discovery that restricts the admissible directed acyclic graph space through protocol-derived structural masks and timeline-aligned indexing. Rather than introducing a new optimization algorithm, we show that explicitly encoding workflow-consistent partial orders reduces structural ambiguity, especially in mixed discrete--continuous panels where within-time orientation is weakly identified. The framework combines workflow-derived admissible-edge constraints, measurement-aligned time indexing and block structure, bootstrap-based uncertainty quantification for lagged total effects, and a dynamic representation supporting intervention queries. In a nationwide annual health screening cohort in Japan with 107,261 individuals and 429,044 person-years, workflow-constrained longitudinal LiNGAM yields temporally consistent within-time substructures and interpretable lagged total effects with explicit uncertainty. Sensitivity analyses using alternative exposure and body-composition definitions preserve the main qualitative patterns. We argue that formalizing workflow-derived constraint classes improves structural interpretability without relying on domain-specific edge specification, providing a reproducible bridge between operational workflows and longitudinal causal discovery under standard identifiability assumptions.

Tadahisa Okuda, Shohei Shimizu, Thong Pham +2

Kyoto University Graduate School of Medicine, Kyoto, Japan · SANKEN, The University of Osaka, Osaka, Japan · Faculty of Data Science, Shiga University, Shiga, Japan +3

Causal DiscoveryCausal Structure LearningTime-Series Causal DiscoveryStructural Causal Models
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Feb 26, 2026

Inferential Mechanics Part 1: Causal Mechanistic Theories of Machine Learning in Chemical Biology with Implications

Machine learning techniques are now routinely encountered in research laboratories across the globe. Impressive progress has been made through ML and AI techniques with regards to large data set processing. This progress has increased the ability of the experimenter to digest data and make novel predictions regarding phenomena of interest. However, machine learning predictors generated from data sets taken from the natural sciences are often treated as black boxes which are used broadly and generally without detailed consideration of the causal structure of the data set of interest. Work has been attempted to bring causality into discussions of machine learning models of natural phenomena; however, a firm and unified theoretical treatment is lacking. This series of three papers explores the union of chemical theory, biological theory, probability theory and causality that will correct current causal flaws of machine learning in the natural sciences. This paper, Part 1 of the series, provides the formal framework of the foundational causal structure of phenomena in chemical biology and is extended to machine learning through the novel concept of focus, defined here as the ability of a machine learning algorithm to narrow down to a hidden underpinning mechanism in large data sets. Initial proof of these principles on a family of Akt inhibitors is also provided. The second paper containing Part 2 will provide a formal exploration of chemical similarity, and Part 3 will present extensive experimental evidence of how hidden causal structures weaken all machine learning in chemical biology. This series serves to establish for chemical biology a new kind of mathematical framework for modeling mechanisms in Nature without the need for the tools of reductionism: inferential mechanics.

Ilya Balabin, Thomas M. Kaiser

Avicenna Biosciences, Inc., Durham, NC, USA

Causal InterpretabilityStructural Causal Models
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Feb 25, 2026

Multi-Level Causal Embeddings

Abstractions of causal models allow for the coarsening of models such that relations of cause and effect are preserved. Whereas abstractions focus on the relation between two models, in this paper we study a framework for causal embeddings which enable multiple detailed models to be mapped into sub-systems of a coarser causal model. We define causal embeddings as a generalization of abstraction, and present a generalized notion of consistency. By defining a multi-resolution marginal problem, we showcase the relevance of causal embeddings for both the statistical marginal problem and the causal marginal problem; furthermore, we illustrate its practical use in merging datasets coming from models with different representations.

Willem Schooltink, Fabio Massimo Zennaro

Department of Informatics, University of Bergen

Structural Causal ModelsCausal AbstractionCausal Representation Learning
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Feb 18, 2026

Causal and Compositional Abstraction

Abstracting from a low level to a more explanatory high level of description, and ideally while preserving causal structure, is fundamental to scientific practice, to causal inference problems, and to robust, efficient and interpretable AI. We present a general account of abstractions between low and high level models as natural transformations, focusing on the case of causal models. This provides a new formalisation of causal abstraction, unifying several notions in the literature, including constructive causal abstraction, Q-ττ consistency, abstractions based on interchange interventions, and distributed' causal abstractions. Our approach is formalised in terms of category theory, and uses the general notion of a compositional model with a given set of queries and semantics in a monoidal, cd- or Markov category; causal models and their queries such as interventions being special cases. We identify two basic notions of abstraction: downward abstractions mapping queries from high to low level; and upward abstractions, mapping concrete queries such as Do-interventions from low to high. Although usually presented as the latter, we show how common causal abstractions may, more fundamentally, be understood in terms of the former. Our approach also leads us to consider a new stronger notion of component-level' abstraction, applying to the individual components of a model. In particular, this yields a novel, strengthened form of constructive causal abstraction at the mechanism-level, for which we prove characterisation results. Finally, we show that abstraction can be generalised to further compositional models, including those with a quantum semantics implemented by quantum circuits, and we take first steps in exploring abstractions between quantum compositional circuit models and high-level classical causal models as a means to explainable quantum AI.

Robin Lorenz, Sean Tull

Quantinuum, London, United Kingdom

Category TheoryCausal AbstractionStructural Causal Models
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Oct 8, 2025

Root Cause Analysis of Outliers in Unknown Cyclic Graphs

We study the propagation of outliers in cyclic causal graphs with linear structural equations, tracing them back to one or several "root cause" nodes. We show that it is possible to identify a short list of potential root causes provided that the perturbation is sufficiently strong and propagates according to the same structural equations as in the normal mode. This shortlist consists of the true root causes together with those of its parents lying on a cycle with the root cause. Notably, our method does not require prior knowledge of the causal graph and yields encouraging results on simulated data and real data from biology and cloud computing.

Daniela Schkoda, Dominik Janzing

Amazon Research Tübingen, Germany · Technical University of Munich, Germany

Structural Causal ModelsRoot Cause Analysis
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May 3, 2025

Causally Fair Node Classification on Non-IID Graph Data

Fair machine learning seeks to identify and mitigate biases in predictions against unfavorable populations characterized by demographic attributes, such as race and gender. Recent research has extended fairness to graph data, such as social networks, but many studies neglect the causal relationships among data instances. This paper addresses a prevalent challenge in many fair machine learning research, which typically assumes independent and identically distributed (IID) data, from the causal perspective. Specifically, this work targets the circumstance where nodes with different neighborhood structures follow different causal mechanisms, violating the invariance assumptions required for classical structural causal models and do-calculus. We base our research on the Network Structural Causal Model (NSCM) framework and develop a Message Passing Variational Autoencoder for Causal Inference (MPVA) to compute interventional distributions for causally fair node classification. We establish theoretical soundness under two conditions: Decomposability and Graph Independence. These conditions formalize when causal mechanism heterogeneity can be overcome by constructing a structural representation that restores invariance and facilitates the computation of interventional distributions using do-calculus in non-IID settings. Empirical evaluations on semi-synthetic and real-world datasets demonstrate that MPVA outperforms conventional methods by effectively approximating interventional distributions and mitigating bias. Our findings demonstrate the potential of causality-based fairness in complex ML applications and motivate future work on relaxing the classic assumptions in algorithmic fairness.

Yucong Dai, Lu Zhang, Yaowei Hu +2

Clemson University · University of Arkansas · Walmart Inc.

Structural Causal ModelsGraph Representation LearningAlgorithmic Fairness
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Feb 27, 2025

Multi-View Causal Discovery without Non-Gaussianity: Identifiability and Algorithms

Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.

Ambroise Heurtebise, Omar Chehab, Pierre Ablin +2

Inria, CEA, University of Paris-Saclay, France · Carnegie Mellon University, Pittsburgh PA, USA · ENSAE, CREST, IP Paris, France +2

Structural Causal ModelsCausal DiscoveryCausal Identifiability
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Jan 31, 2025

What is causal about causal models and representations?

Causal Bayesian networks are often described as causal because they support reasoning about interventions. Yet interventions are defined in the mathematical model, whereas the resulting model predictions concern the effects of actions in the real world. Consequently, whether and how a causal model makes falsifiable predictions about the effects of actions depends on how actions are interpreted as interventions. This requires specifying which actions correspond to which interventions, a step that is frequently left implicit. We develop a formal framework that separates a causal model from the data-generating process it aims to describe and formalizes interpretations connecting actions to interventions. Within this framework, we prove an impossibility result: no interpretation can simultaneously satisfy a collection of desirable properties while remaining non-circular. Under a circular interpretation, every model that fits the observational distribution is interventionally valid, so no experiment can falsify it. We analyze several non-circular alternatives and discuss implications for falsification, causal discovery, causal representation learning, and causal abstraction. More broadly, our framework provides a way to make assumptions about the connections between actions and interventions explicit and to reason formally about the empirical content of causal models.

Frederik Hytting Jørgensen, Luigi Gresele, Sebastian Weichwald

Copenhagen Causality Lab, Department of Mathematical Sciences, and Pioneer Centre for AI, University of Copenhagen, Denmark

Structural Causal ModelsCausal Intervention
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Jul 10, 2024

Towards Complete Causal Explanation with Expert Knowledge

We study the problem of restricting a Markov equivalence class of maximal ancestral graphs (MAGs) to only those MAGs that contain certain edge marks, which we refer to as expert or orientation knowledge. Such a restriction of the Markov equivalence class can be uniquely represented by a restricted essential ancestral graph. Our contributions are several-fold. First, we prove certain properties for the entire Markov equivalence class including a conjecture from Ali et al. (2009). Second, we present several new sound graphical orientation rules for adding orientation knowledge to an essential ancestral graph. We also show that some orientation rules of Zhang (2008b) are not needed for restricting the Markov equivalence class with orientation knowledge. Third, we provide an algorithm for including this orientation knowledge and show that in certain settings the output of our algorithm is a restricted essential ancestral graph. Finally, outside of the specified settings, we provide an algorithm for checking whether a graph is a restricted essential graph and discuss its runtime. This work can be seen as a generalization of Meek (1995) to settings which allow for latent confounding.

Aparajithan Venkateswaran, Emilija Perković

Microsoft · Redmond, WA 98052, USA · Department of Statistics +2

Structural Causal ModelsCausal DiscoveryCausal Structure LearningCausal IdentifiabilityCausal Discovery with Latent Confounding
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May 29, 2024

Invariant Graph Representations for Continuous-Time Dynamic Graphs Under Distribution Shifts

Continuous-Time Dynamic Graphs (CTDGs) enable fine-grained modeling of evolving relational systems. However, most existing CTDG representation learning methods are tailored to in-distribution settings and exhibit limited robustness under out-of-distribution (OOD) shifts. Although recent causal approaches learn invariant representations via interventions, they are primarily designed for static or discrete-time graphs and become computationally prohibitive for CTDGs due to the combinatorial explosion of structural and temporal variations. To address these challenges, we propose CIR, a framework grounded in a novel structural causal model termed the ICCM. To avoid exhaustive interventions, we leverage the Normalized Weighted Geometric Mean (NWGM) to efficiently approximate interventional predictions. We further instantiate ICCM within a practical deep learning architecture that jointly captures invariant structural and temporal patterns through dedicated subgraph extractors, and maintains an environment memory bank to model distributional shifts across evolving contexts. Extensive experiments demonstrate that CIR consistently outperforms existing methods under diverse OOD scenarios.

Lanting Fang, Yulian Yang, Yawei Zhang +3

1Beijing Institue of Technology · Southeast University · Wuhan University

Causal Representation LearningStructural Causal ModelsOOD GeneralizationDynamic Graph LearningInvariant Representation Learning
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