Causal Foundation Models

Causal discovery from observational data is a fundamental yet challenging task in scientific research. While existing approaches are primarily based on conditional independence tests, structure scores, or restrictive functional assumptions, we propose Decoupled Causal Discovery (DCD), a novel decoupling-based perspective that does not rely on these methodologies. DCD directly identifies the Markov boundary (MB) by decoupling non-target variables via weighting functions, such that only variables within the MB preserve dependence with the target under the decoupled distribution. Building on this, DCD iteratively constructs the Completed Partially Directed Acyclic Graph (CPDAG) by exploiting structural asymmetries within the MBs. We establish the theoretical identifiability, soundness, and completeness of DCD. Empirical evaluations demonstrate that DCD achieves strong performance, particularly excelling in challenging noise regimes.

Structural causal models for time series recover contemporaneous and lagged effects, but most methods require complete observation windows and become misspecified when samples are missing. We introduce ITSY, the first continuous-optimization method for causal discovery from irregular time series under a linear model. ITSY reformulates the structural equation so that prediction uses the nearest available history rather than the possibly missing current slice, and jointly imputes missing values while learning both graphs. A weighted reconstruction objective corrects the noise transformation induced by this reformulation. Across synthetic regimes varying missingness, scale, graph density, and noise, and on a real world benchmark, ITSY consistently improves graph recovery over representative SCM-based baselines, demonstrating the effectiveness of the proposed method. The results establish a focused solution for irregular linear first-order dynamics and clarify the assumptions required for nonlinear or higher-order extensions.

Causal discovery from observational data is fundamental to statistics and machine learning, yet determining causal direction without interventions necessitates structural assumptions. Existing identifiability research primarily focuses on continuous variables under additive noise models, often neglecting mixed datasets containing ordinal scales, counts, and continuous measurements. This paper investigates causal discovery in Directed Acyclic Graphs (DAGs) where nodes follow either an ordinal distribution (via an ordered logit model) or a regular one-parameter exponential family distribution. We prove that the edge direction between an ordinal and an exponential family node is distributionally identifiable for generic parameter values. Our findings generalize previous Ordinal-Poisson results to the broader exponential family. Computationally, we introduce a score-based exhaustive search and a masked continuous optimization framework using DAGMA for larger graphs. Numerical results validate the theory, recovering edge orientations within a Markov equivalence class that are unidentifiable under classical structural equation models.

Diagnostic errors and mislabeling are common in biomedicine, which compromise the reliability of predictive models and data-driven outcomes. Stratifying unlabeled biomedical data based on complex relationships between features eliminates the need for data labels and overcomes the limitations of supervised learning. Traditional clustering methods assume restrictive data distributions, making them suboptimal for capturing complex dependencies in high-dimensional biomedical data. This paper introduces a novel cluster-friendly data presentation framework that integrates the non-Gaussian and non-linear feature dependence of copula models with an ensemble of causal structure discovery (CSD) methods based on Directed Acyclic Graphs (DAGs). While copulas model flexible multivariate distributions by relaxing assumptions related to multivariate normality, linear dependence, and symmetric relationships, an ensemble of DAG-based CSD methods identifies stable causal relationships between features. When clustered using K-means, the new data representation obtained by the proposed copula-adapted DAG (CopDAG) ranks first among the 12 methods in normalized clustering accuracy and adjusted Rand index across 16 biomedical datasets. Our CopDAG method predicts ground-truth class labels directly from feature relationships without data annotations and supervised learning, while also providing cluster visualizations and explainable causal structures of the biomedical data features.

Causal discovery aims to uncover causal structures from data and is fundamental to scientific reasoning and intervention-based decision making. Its evaluation relies heavily on structural causal models (SCMs), which specify a causal graph together with the mechanisms that generate data, yet existing studies differ substantially in graph families, mechanisms, and evaluation protocols. The emergence of causal discovery foundation models (CDFMs) further complicates evaluation: performance may reflect not only causal discovery ability, but also overlap between pretraining environments and test SCMs, making results on fixed synthetic benchmarks difficult to interpret. We introduce CausalArena, a unified and evolvable benchmark for causal discovery under a common protocol. Synthetic SCMs supply controlled breadth over structures and mechanisms; semantic operational SCMs provide human-auditable, semantically grounded environments beyond standard synthetic generators; and formula-grounded SCMs test discovery under explicit scientific mechanisms. Public real-world datasets provide an additional external-validity check. Experiments across classical, neural, and pretrained methods reveal substantial ranking shifts across SCM families and protocols, showing that strong performance in one benchmark regime does not reliably transfer to others. These results highlight benchmark diversity and pretraining--evaluation overlap as central challenges for evaluating causal discovery in the foundation model era.

In this paper we study linear non-Gaussian acyclic models (LiNGAM) when used in federated environments. These causal models allow one to go beyond Markov equivalence. However, in many domains data are scarce, and increasing the sample size by centralising data from different clients is not advisable due to regulations such as the GDPR. The federated environment offers an attractive option to balance privacy and causal discovery accuracy. Unfortunately, the standard centralised estimator in the LiNGAM setting, i.e., DirectLiNGAM, cannot be straightforwardly federated. Higher-order cumulant tensors offer a way around this obstacle: they depend only on the joint distribution of the variables involved and add exactly across independent sample groups, so a single communication round suffices in horizontal, vertical, and hybrid partitions. However, FedISHC, i.e., the current federated method along these lines, breaks down under near-symmetric noise. To overcome the above limitation, we introduce the FedRCD family of causal discovery algorithms, and investigate three variants that trade off communication rounds against algebraic noise; two of them are exact federated counterparts of the centralised high-order cumulant (HC) and HC-LiNGAM algorithms, and the single-round variants further effectively support exact unlearning at any granularity, from a single observation to a whole client. Numerical experiments show that at sample sizes typical of real deployments, the entire cumulant-based federated family does not actually rank variables by the population asymmetry that the scores encode at zero. It ranks them by a variance ladder induced by the DAG along its directed paths, the cumulant counterpart of varsortability. Marginal standardisation collapses every cumulant method to near-random ordering, while scale-invariant DirectLiNGAM, not federable under this protocol, is unaffected.

Differentiable causal discovery methods increasingly encode expert priors as forbidden-edge constraints enforced by an Augmented Lagrangian (ALM) penalty, on the assumption that a data-adaptive relaxation mechanism will discount and eventually override a rule the data consistently contradicts. We show this design, which we call \emph{guide, not bind}, fails for two independent, precisely characterized reasons, and that directly repairing both restores it only partially. First, sequential penalty-ramping ALM suppresses a wrongly-forbidden true edge before any counterfactual check can detect it: we give three necessary conditions any adaptive relaxation must satisfy to avoid this (Proposition~\ref{prop:conditions}), prove that DADU---the natural relaxation rule this paper introduces as the object of study---violates all three (Corollary~\ref{cor:dadu_failure}), and confirm the failure across 3{,}072 training runs spanning graphs from 4 to 32 nodes, where a single wrong prior suppresses a true edge in 87--97% of trials under DADU. Second, and independent of any fix to the mechanism, we prove in closed form that the standard correlation-matching objective ties a true edge and its reverse to an identical cost of exactly $2r^2$ (Lemma~\ref{lem:tie}), not because the underlying equal-variance model is unidentifiable, but because normalizing to correlation discards exactly the variance information that would make it identifiable; covariance matching instead separates the two directions by a provable margin of at least $w_0^4$ (Lemma~\ref{lem:separation}).

Causal inference is the practice of estimating the effect of a treatment or intervention from data. It traditionally requires a bespoke pipeline for every new problem: first proposing a causal mechanism, selecting a compatible estimator, and finally training it. Meanwhile, across diverse settings and modalities, much of machine learning has shifted to the paradigm of foundation models: networks pretrained once at scale and applied to new tasks without fine-tuning. Causal foundation models (CFMs) bring this paradigm to causal inference. CFMs are pretrained neural networks that estimate causal quantities, such as the average treatment effect, on entirely new datasets using in-context learning without requiring model updates. This work provides a practical introduction to this emerging area. We summarize the necessary background in causal inference and machine learning before discussing CFMs. Throughout, we include example code and Jupyter notebooks.

Causal discovery aims to uncover the underlying causal relationships given data generated from a system. The goal, however, is not merely to predict causal edges given data, but also to be able to interpret and explain either observed or hypothesized phenomena, such as a particularly large causal effect. We consider this task of conditional causal discovery and cast it as a Bayesian inference problem, in which we target the posterior over causal graphs and parameters conditional on an event such as a causal-effect constraint. Unfortunately, this poses a computational challenge: existing approaches to Bayesian causal discovery struggle when the event has small posterior mass. To address this, we adapt rare-event estimation techniques to perform inference the joint graph-parameter space. Our method gradually drives a particle population toward the constrained region while maintaining samples that approximate the conditional posterior. Empirical evaluation on synthetic graphs validates the accuracy of our approach at small and large scales, and we show in a case study on the Sachs protein dataset how our method can be used to aid scientific exploration by providing pathway-level summaries.

Conditional Independence (CI) tests are the statistical engine of constraint-based causal discovery: in algorithms such as PC (Peter-Clark) and FCI (Fast Causal Inference), skeleton pruning and key orientations follow directly from CI decisions. This survey reviews CI testing with emphasis on assumptions, robustness, and scalability in high-dimensional and mixed-type settings common in biomedical domains. The survey organizes widely used CI methods into six families: partial-correlation, contingency-table, regression, nearest-neighbor, kernel, and machine-learning-based. Special emphasis is provided on the robustness layers that address the limitations of these families. For each family, the survey examines when CI decisions reflect the data-generating distribution and when they fail. By this, we link test-level properties, including power decay with conditioning set size and asymmetric type I/II error consequences, to graph-level errors in skeleton recovery and v-structure orientation. The survey also compares adoption across major R and Python libraries and summarizes open challenges, including mixed-type CI testing without discretization, small-sample error control, and strategies for improving scalability of CI-testing.

Bayesian causal discovery seeks to determine the posterior distribution of causal theories, which are interpreted as directed acyclic graphs (DAGs) that explain the observed data. The resulting posterior allows systematic reasoning regarding epistemic uncertainty within these theories. Nonetheless, finding such graphs is difficult due to identifiability problems and limited observational data. Furthermore, precisely approximating posterior over graphs is challenging given vast range of potential DAGs. Recent Bayesian approaches have addressed some of these challenges, yet they remain limited as they fail to encode dependencies between edges, and lack principled ways to incorporate domain knowledge as inductive biases during the search process. To overcome these limitations, we propose SVI-DAG, a structured variational inference approach to Bayesian causal discovery using observational data and prior beliefs that uses normalizing flows to model dependencies between edges, supporting expressive and multimodal posterior learning over DAGs. To mitigate mode seeking behaviour in evidence lower bound optimization and promote mode coverage, we use stein variational gradient descent to update the node potentials using a kernel in acyclicity space. We evaluate SVI-DAG against 5 state-of-the-art Bayesian DAG learning methods and demonstrate superior performance in uncertainty quantification while remaining competitive in terms of structural accuracy.

Causal Discovery (CD) from observational data faces two fundamental challenges. First, purely statistical methods often lack the power to resolve structural ambiguities in low-sample regimes. Second, although LLM-assisted hybrid approaches improve structure recovery through semantic reasoning, the influence of that reasoning on individual edge decisions remains largely opaque. Consequently, existing hybrid methods fail to satisfy a fundamental requirement: explaining why a particular edge is included or excluded in the learned directed acyclic graph (DAG). This is critical in real-world applications, where no ground-truth DAG exists and every structural decision must be independently justified. We formalize this requirement as decision traceability, requiring every inferred edge to be supported by auditable statistical evidence, Markov Blanket consistency, or explicit domain reasoning. We propose GENESIS, an explainable hybrid CD framework that decomposes graph construction into interpretable decision points. GENESIS first identifies and scores three-node structural motifs, including chains, forks, and colliders, to establish transparent structural priors, then progressively refines the graph by integrating these priors with observational evidence, invoking domain knowledge only when statistical evidence is insufficient. By design, every edge decision is resolved through an auditable source of evidence. Experiments show that GENESIS achieves 100% decision traceability across all settings, establishing explainability as a first-class objective in causal discovery. Despite this additional requirement, GENESIS consistently outperforms purely statistical CD methods on the majority of benchmark datasets across all sample regimes in terms of Structural Hamming Distance (SHD), while achieving performance comparable to state-of-the-art LLM-assisted approaches.

Environmental time-series causal discovery requires expert decisions about method choice, conditional-independence tests, lag horizons, sample-size adequacy, multiple-testing control, and evidence interpretation. Applied inconsistently across datasets, these choices yield graphs that cannot be compared, reproduced, or audited. We present AutoCause, an open-source Python workflow that records each decision, derives defaults from an extended causal-audit module, and admits domain-informed overrides. The workflow wraps four established causal-discovery methods from three families, adds non-causal reference models, and grades links by method-count support. On 145 datasets from DGP-Atlas, TimeGraph, and a topology-derived CausalRivers reference, the methods recover complementary parts of the reference graphs. Majority-supported links are more precise than single-method links on the synthetic benchmarks but not against river topology. AutoCause converts inconsistent expert practice into an auditable, repeatable analysis; causal interpretation remains with the analyst. Available at https://github.com/marcoruizrueda/autocause.

Causal discovery in multivariate time series data is challenging due to complex interactions, high dimensionality, and nonlinear dependencies among variables. Existing methods often struggle to capture these complexities, resulting in inaccurate causal structures. To address this issue, we propose a novel framework that leverages self-attention mechanisms within the transformer architecture for causal discovery. Our approach introduces a novel inverted causal self-attention mechanism (CSAM) that emphasizes latent and indirect causal relationships by inverting tokens and inducing sparsity in attention scores, focusing on significant causal interactions and reducing spurious correlations. Additionally, we develop a global causal algorithm to identify global causal links, providing a holistic metric for causal influence, along with a causal verification module to ensure robustness in the identified causal relationships, enhancing the reliability of our framework. Experiments on both linear and nonlinear datasets, along with ablation studies and sensitivity analyses, show that our framework outperforms existing methods, demonstrating its potential for causal discovery in complex multivariate time series.

Modern telecommunication, cloud, and microservice systems emit correlated alarm cascades when components fail. Root cause analysis (RCA) aims to identify the small set of alarms that initiate each cascade. A common approach learns a causal graph from observational logs and predicts all zero-in-degree alarms in each incident-induced subgraph. However, the learned graph remains fixed and cannot benefit from expert diagnoses of historical incidents. We close this loop with EvoCause. Expert labels constrain which alarms should be source nodes but do not specify the edge edits needed to satisfy those constraints. EvoCause uses a large language model (LLM) to propose semantically plausible graph edits, while deterministic code validates node identities and acyclicity and retains the best graph on a labeled alignment set. At test time, the refined graph alone produces transparent predictions without an LLM call. We also release TeleRCA, an expert-annotated benchmark from a production telecommunication network containing $485{,}681$ alarm events spanning $194$ alarm types over $5{,}621$ resources. On synthetic data, EvoCause initialized with the PC causal discovery algorithm outperforms the unrefined PC baseline, raising Node F1, Case EM, and Graph F1 by $11.59$, $9.40$, and $4.59$ percentage points, respectively, while reducing nSHD by $0.2379$. On TeleRCA, replacing human-readable alarm titles with anonymous identifiers lowers Node F1 and Case EM by $6.12$ and $8.04$ percentage points, respectively, indicating that alarm-name information contributes to graph refinement.

We describe Causal-TS, an open-source Python library for causal discovery in high-dimensional and nonstationary multivariate time series. Causal-TS provides four specialized algorithms-CDNOTS, CDNOTS+, CEDAR, and GRACE-along with wrappers for GES, Granger, LASSO-VAR, and LGES, all sharing a unified conditional independence (CI) test layer with GPU acceleration via PyTorch. A regime discovery pipeline detects structural breaks via pluggable changepoint detectors and runs discovery per regime with regime-specific parameters. A command-line interface, synthetic data generators, and optional DoWhy integration provide an end-to-end pipeline from raw time series to causal effect estimates. The library is pip-installable, tested on Python 3.10--3.12, and available at https://github.com/bloomberg/causal-ts.

Learning causal graphs from interventional data is a challenging problem with broad applications. In molecular biology, for example, a central goal is to uncover gene regulatory networks from large-scale perturbation data. An ideal algorithm for this task should scale to thousands of nodes, incorporate interventions even when their targets are unknown, quantify uncertainty, and provide identifiability guarantees. However, existing approaches---e.g. approaches using score-based optimization or approximate Bayesian inference---often fail to meet all of these criteria. To address these limitations, we develop Amortized Bayesian Causal Discovery of Extended Factor Graphs (ABCDEFG). Our method guarantees exact acyclicity, scales to graphs with thousands of nodes, and naturally handles interventions even when their targets are unknown. Additionally, ABCDEFG estimates a posterior distribution whose maximum a posteriori estimate provably identifies the true causal graph up to an equivalence class. On simulated datasets, ABCDEFG achieves state-of-the-art accuracy, producing a well-calibrated posterior distribution while outperforming previous score-based and approximate Bayesian methods. Applied to large-scale single-cell perturbation data, ABCDEFG identifies both established and novel gene targets of growth factors.

Discovering the direct causes and effects of a target variable from observational data is a fundamental problem in causal discovery, with broad applications in domains such as gene regulatory analysis and biomedical research. Existing causal discovery methods either learn a global causal structure, which incurs substantial computational cost, or assume the absence of latent variables and selection bias, assumptions that are often violated in real-world settings. Motivated by these challenges, we study local causal structure learning in the presence of latent variables and selection bias. Specifically, we first characterize a local region that enables target-specific causal discovery without recovering the entire global structure. We then establish a theoretical bridge between causal information learned from the observed distribution induced on this local region and the corresponding information in the global causal structure. Building on these foundations, we propose LoCaLS, a local causal structure learning algorithm that is sound and complete under standard assumptions and identifies the same direct causes and effects of a target variable as those identifiable by global causal discovery methods, while allowing for latent variables and selection bias. Extensive experiments on random and real-world structures demonstrate that the proposed method consistently achieves higher structural accuracy than existing local methods while requiring substantially less computational effort than state-of-the-art global methods. Furthermore, applications to two real-world gene expression datasets reveal biologically plausible target-specific causal structures, demonstrating its practical applicability in large-scale biological data analysis.

Causal discovery methods have shown strong performance in temporal systems, but they typically rely on regular and discrete lag structures, limiting their applicability to regularly sampled data. However, many real-world tasks require dealing with irregularly sampled streams of events, such as sensor streams, healthcare data, and financial transactions. In this work, we propose an extension of PCMCI+, a state-of-the-art method for causal discovery on regular multivariate time series, to allow for handling irregular time series. Instead of modelling causal relations through fixed-lag dependencies, our method aggregates causal influence over predefined temporal windows. We evaluate our method on synthetic irregular event streams with known causal structures under different signal-to-noise ratios, showing that it consistently recovers the underlying causal graph and substantially outperforms the standard PCMCI+ on irregularly sampled data.

Identifying root causes in production microservice failures requires reasoning over large-scale, multimodal telemetry spanning metrics, logs, and traces, a problem that has proved resistant to both classical and LLM-based approaches. The OpenRCA dataset exemplifies these challenges: it is large-scale, multimodal, and lacks detailed domain knowledge, and yields consistently low accuracy across all existing methods. We show that classical causal discovery methods and existing LLM-based multi-agent systems fail to reliably identify root causes on this benchmark, and present a Structured Multi-Agent RCA pipeline that substantially outperforms existing LLM-based and classical baselines, supporting both domain-knowledge and knowledge-free operating modes. To diagnose where failures originate, we introduce a reverse reasoning agent that, given the correct answer, identifies which signals in the extracted anomalies support it and determines whether Stage~1 had access to those signals, classifying each failure as Reasoning Gap (evidence present but unused) or Data Ambiguity (evidence genuinely absent). This analysis reveals that the required evidence is present in the vast majority of failures: the bottleneck is not data access but the agent's ability to reason over it correctly. We further introduce an automated rule mining pipeline that systematically extracts discrimination rules from reverse reasoning reports, reducing reliance on manual knowledge curation. Across all configurations, model reasoning capability and domain knowledge are the primary constraints: stronger models embed more domain expertise, and explicit knowledge injection partially compensates for this gap. Reasoning performance remains practically bounded even when evidence extraction is perfect: scaffold engineering and better data pipelines alone cannot close this gap; progress requires improvements at the model level.

Causal discovery algorithms learn a network that describes the causal dependencies among random variables. A common workflow involves first utilizing conditional independence properties on observational data to determine partially directed causal relationships, then applying interventions to orient the unknown causal directions. A critical assumption for the first step is faithfulness: a requirement that causally linked variables exhibit statistical dependence. Many natural systems include buffering and stabilizing pathways that cancel out to achieve systemic robustness. This cancellation of pathways violates faithfulness, leading causal discovery algorithms to incorrectly remove causal dependencies. In this paper, we argue that hard interventions contain information about the presence/absence of causal linkage that is overlooked in the first stage of structure discovery. We show that a mild assumption -- called intervention-immediacy faithfulness -- that allows cancellations, is sufficient to nonparametrically identify causal structures with hard interventions. These results position interventions as the primary carriers of information about causal structure, which should take precedence over conditional independence testing. To flip the paradigm, we also specify equivalence classes when the identification criteria are not met due to limitations in the scope of interventions.

Causal discovery from observational tabular data remains fundamentally challenging, primarily due to the heterogeneity of underlying causal mechanisms and the high-dimensional combinatorial search space of Directed Acyclic Graphs (DAGs). In this paper, we propose \textbf{DAG-FM}, a novel foundation model architecture that amortizes causal discovery. Unlike direct matrix prediction, DAG-FM decomposes the causal discovery process into two auto-regressive stages using two specialized Transformer-based sub-modules: a leaf-node predictor and a parent-node predictor. To effectively model complex row-column interactions, we adopt a robust tabular interaction block to output feature-wise representations. Crucially, to handle diverse and unknown Functional Causal Model (FCM) assumptions in real-world scenarios, we introduce Mixture-of-Leaf-Experts (MoLE), allowing the model to dynamically route and adapt to identifiable mechanism families. Through an iterative inference algorithm, DAG-FM seamlessly extracts causal orderings and constructs valid DAGs. Extensive experiments demonstrate that DAG-FM achieves state-of-the-art performance on both synthetic benchmarks and complex real-world datasets, significantly outperforming traditional classical algorithms and recent foundation models in both accuracy and scalability.

Causal discovery, the process of recovering underlying causal structures from observational data, is a fundamental pursuit across scientific disciplines. Over the past decades, numerous algorithms have been developed to tackle this challenge through workflows tailored to the specific causal mechanisms underlying each type of dataset, demonstrating effectiveness across a wide range of applications. However, as the volume and heterogeneity of real-world data continue to grow, this dataset-specific approach inevitably leads to a fragmented, test-driven paradigm that struggles to scale to the demands of modern scientific discovery. To address this, we formulate the Causal Discovery Foundation Model (CDFM) as a unified, general-purpose framework for zero-shot structural inference. To ensure reliable generalization across unknown domains, we first investigate the theoretical boundaries of causal identifiability, revealing the indispensable role of causal prior mechanisms in this process. Building on these insights, we formulate a principled variational framework that treats unknown causal mechanisms as latent variables and mathematically decomposes the intractable marginal likelihood into distinct, tractable learning modules. The variational decomposition provides a conceptual design principle for the architecture design of CDFM, while comprehensive causal knowledge guides the large-scale synthesis of our pretraining data. By pretraining on a massive, highly diverse space of synthetic structural causal models, CDFM successfully internalizes complex statistical asymmetries. Extensive experiments demonstrate that CDFM consistently outperforms traditional algorithms, driving a paradigm shift toward a general-purpose causal discovery foundation model.

Expert background knowledge is often available in practical applications of causal discovery. Such constraints on the true causal graph can help causal discovery in terms of identifiability of causal effects and accuracy of the learned structure, but also in reducing the space of candidate causal graphs. As causal discovery can become computationally expensive for large number of variables, it is crucial to utilize background knowledge effectively during the causal discovery process. However, most current methods only use background knowledge in a postprocessing step after causal discovery to refine the learned graph. In this work, we develop a framework for utilizing background knowledge during the causal discovery process, focusing especially on scalable causal discovery methods that recover only a subset of the whole graph. We implement our framework for multiple algorithms and empirically show that utilizing background knowledge can both reduce computational requirements and increase the quality of the learned structures.

Bayesian causal discovery is widely used for its ability to quantify epistemic uncertainty over directed acyclic graphs (DAGs) through posterior inference. However, its behaviour under latent confounding remains poorly understood, as existing work typically notes that confounding breaks identifiability without characterising how the posterior distribution over DAGs responds. In this work, we analyse posterior behaviour under latent confounding in linear Gaussian causal models, focusing on additive latent confounding between exactly two observed variables. We derive a critical correlation threshold above which the score function favours graphs with a spurious edge between the confounded variables, and show that this threshold decreases with sample size -- more data lowers the correlation required for the spurious edge to be favoured. Beyond this threshold, we characterize two distinct posterior failure regimes determined by the local structure around the confounded variables. Our findings are supported by exact posterior computations on multiple graph structures, demonstrating both the predicted failure regimes.

Causal discovery from unstructured data is a challenging yet underexplored task in high-expertise domains such as healthcare, finance, and education. Existing methods typically leverage the general knowledge of large language models (LLMs) to identify causal factors from unstructured data and annotate them into structured data for causal graph construction. However, they remain limited by two key challenges (CHs): (CH1) insufficient identification of latent factors, which are implicit in the data yet essential for causal discovery, due to the lack of domain-specific knowledge; and (CH2) unreliable factor annotation, caused by the lack of domain-grounded reasoning, which propagates errors to the resulting causal graphs. To address these challenges, we introduce a novel Domain Knowledge-enhanced Causal Discovery framework (DKCD) for causal discovery from unstructured data in high-expertise domains with three interconnected components: (1) Knowledge Mining: It retrieves relevant domain knowledge based on observable factors to support subsequent causal reasoning. (2) Knowledge-guided Causal Reasoning: Reasoning with relevant knowledge, it discovers latent causal factors to address CH1 and generates key causal clues for more accurate data annotation to address CH2. (3) Causal Structure Discovery: It constructs the final causal graphs based on a more complete factor set and accurate annotations. Experiments on two domain-specific datasets show that DKCD significantly improves both causal factor identification and causal graph construction.

The operational integrity of complex industrial systems relies on precise anomaly detection and diagnosis. The vast majority of existing methods narrowly focus on capturing temporal similarities of representations, often overlooking the disruption of internal causal relationships, which characterizes system failures and latent anomalies. In this paper, we propose a novel framework (CAAD) that reframes anomaly detection as the continuous verification of Granger causality consistency through exogenous variables. Specifically, the CAAD framework models exogenous time-series variables as residuals, identifying anomalies as significant deviations caused by external interventions. The proposed framework leverages multi-scale alignment to internalize system dynamics and utilizes a gradient-based matrix to monitor internal causal relationship breakdowns. By quantifying causal deviations of both dynamic evolution and relational topology, the CAAD is able to capture subtle causal shifts to achieve precise anomaly detection. Extensive experiments on real-world industrial datasets demonstrate that the CAAD achieves high-precision anomaly detection, outperforming most state-of-the-art baselines.

Biological systems exhibit a hierarchical structure, characterised by directed flow from upstream regulators to downstream effects. Although this ordering provides a natural scaffold for causal inference, most causal discovery and GRN methods either ignore the tiered organisation or condition on all upstream variables, which becomes infeasible for high-dimensional omics data. We present ASCEND (Ancestral Scalable Causal discovEry via iNherited Descent), a constraint-based framework that leverages known two-tiered structure to enable genome-scale causal discovery. ASCEND introduces a divide-and-conquer strategy that maintains dynamically updated ancestral conditioning sets for each downstream variable, dramatically reducing the number of conditional independence tests required, and achieves polynomial-time complexity where traditional approaches face exponential blow-up. Through extensive simulations and real biological data, we demonstrate that ASCEND accurately recovers ancestral relationships, scales properly and much faster, and outperforms existing gene regulatory network inference methods in both causal precision and computational efficiency. The algorithm's ability to resolve directionality makes it particularly suited for integrating multi-omic data where upstream regulators (e.g., SNPs, methylation sites) and downstream responses (e.g., gene expression) are measured jointly.

Local causal discovery is a scalable alternative to global structure learning. However, it can struggle to identify valid adjustment sets in data-scarce settings because of finite-sample uncertainty, incomplete local neighborhoods, and unresolved Markov equivalence. Although many application domains provide structured background knowledge, its integration into local causal discovery remains limited. We propose b-LOAD, a knowledge-informed extension of the LOAD algorithm for local discovery of optimal adjustment sets. b-LOAD incorporates prior edge constraints directly into the local structure-learning procedure and uses Meek's rules to expand the discovery frontier dynamically, yielding a knowledge-constrained partially directed graph over the relevant local subgraph. This strategy helps prevent structurally relevant nodes introduced by prior knowledge from being excluded by local search. We prove that, under sound background knowledge, the procedure monotonically refines the admissible equivalence class and can enlarge the set of identifiable causal queries, enabling recovery of optimal adjustment sets that are not identifiable from observational conditional-independence information alone. Empirically, b-LOAD improves downstream causal effect estimation relative to purely data-driven and standard knowledge-augmented baselines, particularly in data-scarce and structurally complex regimes. Results on real-world biological networks show that locally targeted prior knowledge provides the largest gains and remains beneficial under moderate structural noise. These findings position b-LOAD as a scalable approach for converting fragmented domain knowledge into more reliable causal-effect estimation.

Causal discovery with nonlinear mechanisms and latent confounders remains challenging. Existing methods often rely on either linear assumptions or causal sufficiency, limiting their applicability. We propose an MDL-based causal discovery framework that explicitly accounts for latent confounders while allowing flexible nonlinear mechanisms by minimizing the luckiness normalized maximum likelihood (LNML) code-length. The causal relationship between each variable pair is determined by selecting the shortest code-length of the causal model, and we introduce the notion of $Δ$-pseudo-collinearity to identify dependencies induced by latent confounders. Based on these ideas, we develop a greedy algorithm, termed Pseudo-Collinearity Guided Causal Discovery (PCG-CD). Experiments on synthetic and real-world datasets demonstrate that the proposed method accurately recovers directed causal relationships and effectively detects latent confounders.

Causality has become an increasingly important tool for gaining a deeper understanding of complex systems. Among various causal analysis methods, causal discovery, which identifies causal relationships among variables from data, has been widely used to uncover underlying causality in diverse processes. However, while multistage processes are prevalent in many fields, existing causal discovery methods may produce counterintuitive results, given the known process knowledge, and may not be computationally efficient for handling large datasets typical of multistage processes. To address this gap, we propose a novel causal discovery method called Order-based Causal Discovery for Multistage Processes (OCDM). OCDM is designed to infer the causal structure of multistage data while preserving their inherent hierarchical and sequential structure by explicitly incorporating process knowledge into the causal discovery process. Specifically, we propose a structural knowledge-informed order-inferring algorithm that infers the causal order of variables by incorporating information about the stage from which each variable originates, based on an order-based causal discovery framework naturally suited for inherently ordered multistage data. Furthermore, to eliminate spurious edges from the initial causal graph generated based on the inferred causal order, we introduce a novel pruning technique using stochastic gated neural networks, which offers greater computational efficiency compared to existing methods. Through experiments on various datasets, we demonstrate that OCDM effectively infers the causal structure of multistage processes, outperforming existing methods.

Learning causal models from high-dimensional data is a significant challenge, particularly in real-world settings where violations of core assumptions lead to causal identifiability issues. Although massive amounts of prior knowledge are available, and contain valuable causal information, effectively integrating this knowledge into the causal discovery process remains an open problem. We introduce CausalSTeward (CAST), a novel human-in-the-loop framework for interactively assembling large causal models. CausalSteward is a multi-agent collaborative system that tackles high-dimensional causality through a divide-and-conquer approach where large clusters of variables are iteratively partitioned and then separately analyzed. Our framework fuses prior knowledge with a data-driven approach by using tailored tools such as retrieval augmented generation and conditional independence tests. Finally, we use this work to examine the capabilities and limitations of causal reasoning in multi-agent frameworks, and how the human-in-the-loop can contribute to accurate and trustworthy results.

Causally linking disease-related factors to image-derived biomarkers provides a powerful pathway to understanding disease mechanisms. Despite growing interest in applying causal artificial intelligence (AI) approaches for this task, these methods still need to be adapted for complex medical images, and especially, neuroimaging. However, the lack of ground-truth data presents a barrier to development. To bridge this gap, we developed and tested a method for generating synthetic neuroimages, which adhere to a user-specified causal structure describing the non-image to image variable relationships, permitting the creation of ground-truth neuroimaging datasets. In the simulated T1-weighted magnetic resonance images, anatomical variability is modeled by sampling from a subspace estimated from real data and deforming a template image to create unique simulated subjects. Causal relationships are encoded via precise volumetric changes of any region-of-interest without unwanted global artifacts. We achieved relative volume errors of 0.3-2.66% for the targeted regions-of-interest and demonstrate their statistically significant causal relationships, while maintaining mean absolute errors for non-target brain regions between 0.034-0.397ml. An initial evaluation of causal discovery methods exposes their limited ability to suppress spurious connections, highlighting the need for image-appropriate methods. Our framework is the first to enable the generation of realistic synthetic 3D neuroimages with explicit causal control that can serve as the missing ground-truth data necessary for the objective benchmarking and development of causal AI methods.

We introduce TabPFN-CFM, a causal foundation model that can handle multiple causal problems. TabPFN-CFM predicts both causal structure and outcomes from observational data, supports queries on all three levels of Pearl's Causal Hierarchy and uses known graph structure when available to improve predictions. TabPFN-CFM is trained on synthetic datasets, and generalises to real datasets, demonstrating improved performance over both structural and outcome prediction baselines.

From climate teleconnections to gene regulation, modern time-series datasets encompass tens or hundreds of interacting variables, making causal discovery increasingly challenging. Constraint-based methods offer statistical rigor but their nonlinear CI tests are infeasible at scale, while score-based alternatives avoid CI testing but require arbitrary thresholds to binarize continuous edge scores. We propose GRACE ($\textbf{G}$ated $\textbf{R}$efinement for $\textbf{A}$ccurate $\textbf{C}$ausal $\textbf{E}$dge discovery), which refines constraint-based discovery using Hard Concrete gates with $L_0$ regularization: each candidate edge has an independent gate whose values concentrate near 0 or 1, yielding a clean bimodal separation that makes the binary decision robust, unlike the narrow, overlapping score distributions produced by $L_1$ and attention-based methods. A fast linear CI skeleton provides high-recall candidates; a single gated model then prunes false positives by learning which edges genuinely improve prediction, with automatic regularization adapted to problem dimensions and skeleton density. Systematic experiments on synthetic benchmarks, spanning diverse graph topologies (scale-free, Erdős-R'enyi, small-world) and dimensionalities up to $d=100$, show that GRACE substantially improves F1 over its base CI method while maintaining high precision, and outperforms attention-based and score-based alternatives. GRACE matches or exceeds expensive nonlinear CI tests at a fraction of the cost ($75\times$ faster). On a real-world river flow dataset, where rainfall confounders, variable propagation lags, and distributional shifts violate standard assumptions, a temporal bootstrap variant of GRACE recovers 9 of 11 causal edges along the Elbe River with only 1 false positive ($F_1 = 0.86$, AUROC${} = 0.99$), reducing the skeleton's 106 false positives by 99%.

Recent attempts to combine large language models (LLMs) with causal discovery ask models to infer pairwise directions, propose graph structures, or inject language-model outputs as priors and constraints. These approaches promise faster analysis, but they also obscure whether a causal evidence is supported by data and assumptions or by textual associations, prompt artifacts and hallucinated mechanisms. We argue for a different role for agents in causal discovery. Agents should inspect data, retrieve context, explain method assumptions and clarify graph outputs, but they should not supply edges, orientations, priors, constraints or causal conclusions. We propose the principle that agents assist the workflow, while causal claims remain grounded in data, explicit assumptions, formal algorithms, diagnostics and user or domain-expert decisions. We instantiate this principle in causal-learn+, an online platform that coordinates data analysis, preprocessing, method recommendation, expert-knowledge incorporation, formal discovery and interpretation around the algorithmic ecosystem of causal-learn. A case study on Big Five personality data illustrates agent-assisted pipeline of causal discovery without turning language-model unreliability into causal evidence. The platform is available at causallearn.com.

Causal reasoning, which encompasses the discovery of causal structures and the inference of causal effects, is fundamental to data-driven decision making. In practice, data for reliable causal analysis are often distributed across institutions and cannot be centralized due to privacy regulations or communication constraints. Federated learning (FL) addresses this by enabling collaborative analysis without raw data sharing, giving rise to the rapidly growing field of federated causal discovery (FCD) and inference (FCI). However, the interdisciplinary nature of this field and the absence of a comprehensive survey present barriers to entry for researchers. This paper bridges that gap by providing a systematic review through multi-dimensional taxonomies. Grounded in the three core design decisions underlying any FCD solution, namely how structures are learned, how data are partitioned, and what structural knowledge each party obtains, we organize FCD along three axes: methodological paradigm, federation topology, and structural scope. We further examine key practical dimensions, including temporal dynamics, data heterogeneity, missing data, and non-identical variable sets. For FCI, we categorize methods by target estimand (average versus individualized/conditional treatment effects) and by estimation strategy, from classical weighting methods to modern deep generative architectures. Unlike prior works that treat FCD and FCI separately, we formalize their connection as complementary stages of a unified federated causal reasoning pipeline, where FCD supplies the structural knowledge required for valid effect estimation in FCI. Finally, we highlight their shared concerns regarding privacy, communication efficiency, theoretical guarantees, and application domains, and conclude by identifying open challenges for future research.

Causal discovery is critical for understanding complex data-generating mechanisms, yet traditional algorithms often struggle with highly non-linear and noisy systems, or suffer from severe computational bottlenecks. Recent tabular foundation models based on Prior-Data Fitted Networks (PFNs) have demonstrated remarkable zero-shot inference capabilities, but their potential for explicit structural causal discovery remains underexplored. To bridge this gap, we propose DCD-PFN, a decoupling-aware foundation model for causal discovery. Instead of directly amortizing global graph reconstruction, DCD-PFN focuses on local causal discovery through a decoupling-based paradigm. Through pre-training on diverse synthetic Structural Causal Models (SCMs), the model learns sample-wise decoupling weights that enable Markov boundary (MB) identification. Furthermore, by leveraging parallelized local discovery, DCD-PFN efficiently reconstructs global causal graphs while remaining grounded in the theoretical foundations of decoupling-based causal discovery. Experiments demonstrate that our foundation model achieves robust zero-shot generalization.

Estimating causal effects in industrial time series requires handling temporal dynamics, time-varying treatments, and unobserved confounders. Existing causal foundation models (CausalPFN, CausalFM) operate only on static cross-sectional data; neural temporal methods (CRN, G-Net) require per-dataset training; and concurrent temporal-PFN proposals have not been demonstrated at industrial scale. None output explicit per-pair reliability signals alongside their CATE estimates. We introduce Temporal Causal Prior-Data Fitted Networks (TCPFN), a foundation model for zero-shot temporal causal discovery with learned reliability signals. TCPFN makes four contributions: (1) a Causal Judgment Head that jointly predicts null-effect probability, confounding strength, identifiability, mediation fraction, and causal regime; (2) a mixed training prior covering six causal regimes (independent, direct, confounded, mediated, time-varying confounded, feedback) plus CausalFM-style front-door and instrumental-variable priors; (3) a discrete-token panel-data architecture with cross-attention masking that prevents inter-horizon leakage; (4) zero-shot inference at industrial scale via FAISS-based context selection and one-step posterior correction. On 19 benchmark datasets across five domains, TCPFN achieves competitive zero-shot causal discovery: AUROC 0.96 on Tennessee Eastman, 0.93 on SWaT, 0.98 on Causal Rivers, 0.97 on CAUSRCA. The null detector reaches NullF1 0.94, AUROC 0.99. TCPFN scales to V=1,275 on a proprietary Kraft pulp-and-paper dataset in 6 hours on a single GPU; PCMCI, a CPU-only library, on a V=666 sub-panel of the same data took 81.5 hours, extrapolating by O(V^2) to ~12.5 days at V=1,275. TCPFN's top edges identify cross-subsystem causal relationships while PCMCI's surface within-instrument controller-measurement coupling -- a scalability case study.

Recent work on Kan-Do-Calculus (KDC) has established that the boundary between passive observation and active intervention in causal inference is a category-theoretic bi-adjunction, with interventions modeled by left Kan extensions and conditioning by right Kan extensions. This paper introduces two causal discovery algorithms under latent confounding, building on the information-geometric and categorical consequences of KDC. In smooth statistical settings, Radon-Nikodym derivatives between observational and interventional measures induce local causal vector fields; failures of these fields to close under Lie brackets become computable Frobenius residuals, which we interpret as witnesses of failed visible integrability and possible latent or unmodeled structure. Our first algorithm, BRIDGE (Bracket Residuals for Interventional Discovery and Geometric Estimation), combines an interventional density or Radon-Nikodym-ratio engine with a geometric screen that proposes a high-recall family of admissible arrows, identifies non-closing visible pairs as latent-obstruction candidates, and passes the reduced family to downstream score-based or differentiable discovery routines. The second algorithmic contribution, Spectral Kan-Do Flow Matching (SKFM), learns amortized intervention fields and factors latent curvature spectrally, exposing the direct Lie-space endpoint toward which BRIDGE points. A detailed set of experiments show that both algorithms are capable of discovering causal models with latent confounders while collapsing the super-exponential space of possible DAGs by many orders of magnitude. This paper introduces a new paradigm in causal discovery, where latent structure is inferred directly from the geometry of intervention-induced flows.

Causal abstractions formalize when a high-level structural causal model (SCM) captures the interventional behavior of a lower-level SCM. Existing applications of this notion largely follow a hypothesis-testing paradigm: an expert proposes a candidate high-level model and then evaluates if the low-level system implements it. We study the complementary problem of learning a high-level model directly from low-level measurements. Our contributions leverage hypotheses from low-rank causal discovery, and can be summarized as follows: (1) we show that observations generated by a low-rank graph induce latents that form a causal abstraction, (2) we provide identifiability results about these latents, and (3) we propose a practical objective to learn this high-level SCM.

Causal discovery methods commonly assume that all data is independently and identically distributed (i.i.d.) and that there are no unmeasured variables affecting the system. In practice, these assumptions are often violated, leading to inaccurate inference. In this paper, we study how to identify hidden confounding and selection biases from causal mechanism shifts. In particular, we show that structural biases lead to dependent mechanism shifts. That is, by considering for which variables the mechanisms change given data from different environments, we can tell which variables are unbiased, which are subject to hidden confounding, and which are undergoing selection bias. We formalize this into an empirically testable criterion based on mutual information, and show under which conditions it identifies structural biases. To tell which nodes are subject to what kind of bias, we introduce the StruBI algorithm. Experiments on synthetic and real-world data show that StruBI works well in practice, accurately recovering affected variable sets and types of biases, outperforming the state-of-the-art by a wide margin.

Objective sleep assessment relies on polysomnography (PSG), yet clinical impact is often better reflected in patient-reported outcomes (PROs) such as sleepiness and fatigue. Existing summary indices, including the Apnea-Hypopnea Index (AHI), provide limited insight into the multidomain physiology underlying functional recovery. We propose an interpretable, causal-discovery-guided framework for deriving a hierarchical Sleep Recovery Score (SRS) from multimodal PSG. Using two large population cohorts (MESA: $n=1{,}540$; MrOS: $n=825$), we apply directed acyclic graph (DAG) learning to identify candidate physiological drivers spanning respiratory burden, hypoxic burden, sleep fragmentation, sleep architecture, and autonomic regulation. Although derived from clinical PSG, these domains map naturally to sensing streams increasingly available in connected health technologies, including wearable ECG, oximetry, and sleep-stage estimation devices. To preserve mechanistic plausibility, we introduce a two-stage screening process that combines physiology-based constraints with constrained LLM-assisted auditing to identify and remove structural confounders and construct-overlapping variables. Across cohorts, these five domains emerge as recurrent physiological domains associated with recovery, and the resulting SRS shows up to $3.4\times$ stronger alignment with perceived recovery than AHI. By linking multimodal sleep physiology to patient-centered outcomes through an interpretable, bias-aware, and domain-structured framework, this work provides a practical foundation for recovery modeling across both clinical sleep studies and emerging smart and connected health settings.

Causal discovery seeks to uncover the causal dependencies among variables. For this purpose, we propose an algorithm called Tensor-based Second-order Causal Discovery (TSCD). Its input is a tensor obtained from the covariance matrices of observational and interventional data. Assuming the causal dependencies follow a linear structural equation model on a directed acyclic graph (DAG), TSCD outputs the DAG and the functions on its edges, requiring only that the noise variables are uncorrelated. We also implement a version of the approach for nonlinear models. Our focus on second-order statistics (via the covariance matrices) is motivated by their statistical and computational efficiency relative to higher-order moments, their identifiability relative to first-order statistics, and that they work regardless of whether the variables are Gaussian. We show that TSCD has identifiable causal order and parameters from a number of interventions that is logarithmic in the number of variables. Experiments show that TSCD is robust to noise, competitive with existing methods, and scales to hundreds of variables.

Constraint-based causal discovery relies on repeated conditional independence tests, but fast nonparametric tests often sacrifice calibration, especially when variables depend on the conditioning set through nonlinear relationships. We introduce BLITZ (Broad-to-Local Independence Testing via residualiZation), a nonparametric conditional independence test designed to run well under a second while maintaining the accuracy needed for the thousands of queries performed by constraint-based causal discovery algorithms. BLITZ first removes broad smooth dependence on the conditioning set using low-order polynomial regression, then applies a small nonlinear feature map and residualizes those features with shallow tree regressions. The resulting statistic tests residual cross-covariance, with a moment-matched chi-square approximation to the null distribution. We show theoretically that the two-stage design reduces the effective complexity faced by the tree residualizers, allowing shallow trees to control residual conditional-mean bias while avoiding excessive overfitting. In simulations, BLITZ provides better null calibration than fast kernel, random-feature, and regression-based competitors while remaining among the fastest methods tested. In causal discovery experiments on synthetic graphs and flow-cytometry data, BLITZ yields more reliable endpoint orientations among retained adjacencies and competitive structural recovery. These results suggest that broad-to-local residualization is a practical route to calibrated, scalable nonparametric conditional independence testing for causal discovery.

Causal discovery from observational data remains challenging due to the need to recover directed structure and latent confounding without interventions. We propose FoundCause, an amortized causal discovery model trained entirely on synthetic data that maps datasets directly to causal graphs in a single forward pass. By learning from large collections of simulated structural causal models, FoundCause captures transferable statistical patterns that generalize beyond individual datasets. The architecture incorporates several key inductive biases for causal discovery. It uses a permutation-invariant transformer encoder with alternating attention over samples and variables to jointly model cross-variable dependence and per-variable distributions. Pairwise statistical features derived from classical asymmetry measures are injected through statistics-conditioned attention, guiding the model toward known causal signals. A factorized decoder separates edge existence from direction, while a triangular refinement module enables reasoning over higher-order causal motifs such as chains and colliders. In addition, a dedicated confounder module based on learnable latent tokens explicitly models hidden common causes, and the model explicitly handles missing data via its masked input representation. To our knowledge, FoundCause is the first amortized causal discovery approach to explicitly model latent confounding. FoundCause outperforms 11 classical non-amortized methods (e.g., PC, GES, NOTEARS-style optimization) and 4 amortized causal discovery methods on 15 real-world datasets, achieving +9.6% improvement in $F_1$, +1.2% in AUROC, and an 18.9% reduction in structural Hamming distance relative to the strongest non-amortized methods, while performing inference in a single forward pass.

Granger Causal Discovery (GCD) is fundamental for analyzing temporal dependencies in complex systems. However, existing neural GCD methods predominantly rely on a "one-size-fits-all" paradigm, struggling to capture distribution shifts and dynamic regime changes inherent in real-world time series. This often leads to entangled representations and spurious causal graphs. In this paper, we propose CausalMoE, a billion-scale multimodal Granger causal foundation model that explicitly models patch-level heterogeneity. CausalMoE introduces a Pattern-Routed Mixture of Heterogeneous Experts, which dynamically identifies latent temporal patterns and routes patches to specialized domain experts, effectively decoupling regime-specific mechanisms from shared dynamics. To ensure interpretable graph recovery, we design a Causality-Aware Self-Attention mechanism operating across variables, yielding sparse Granger causal graphs via proximal optimization. Furthermore, CausalMoE is the first to integrate LLMs and VLMs to align numerical signals with textual and visual priors, regularizing causal estimation in complex scenarios. Extensive experiments demonstrate that CausalMoE establishes a new state-of-the-art on fully supervised benchmarks, while effectively generalizing to few-shot settings where traditional methods fail.

Causal discovery aims to uncover causal structures from observational data, which is crucial for real-world decision-making. However, different causal discovery algorithms can produce divergent results that conflict with each other, complicating the identification of accurate causal graphs. Traditional approaches rely on numerical values and statistical assumptions, often ignoring rich domain-specific information, such as feature descriptions, which could also help structure learning. While recent works explore using Large Language Models (LLMs) to infer causal relations via direct queries, such methods can be unreliable due to a lack of alignment with the actual data. To address these limitations, we propose Causal Ensemble Agent (CEA), a novel framework that aggregates structural insights from statistical discovery experts across different graph levels via linear opinion pooling, and uses an LLM as a meta-referee to dynamically reweight experts when the aggregated confidence is close to the decision boundary, thereby composing an improved and more complete causal graph. Extensive experiments on both synthetic and real-world datasets demonstrate that CEA achieves the strongest overall performance across a wide range of causal discovery methods, highlighting the effectiveness of using LLMs for meta-analysis in causal discovery.

We study causal discovery from observational data when some variables are hidden and the data-generating process follows a location-scale noise model (LSNM). Existing methods that handle hidden confounders typically assume additive noise, but in practice, causes often modulate not just the mean but also the variance of their effects. We prove that acyclic directed mixed graphs (ADMGs) satisfying a bow-free condition are identifiable under LSNM with hidden variables, establishing the first identifiability result for causally insufficient models beyond noise additivity. We further provide sufficient conditions for identifying causal direction even when the bow-free assumption is violated. Our two-stage algorithm, LSNM-UV, is sound and complete, and experiments demonstrate improved performance over additive baselines on heteroscedastic data.

Modern image-analysis pipelines often convert images into structured semantic variables, such as facial attributes, object concepts, and scene descriptors. Learning directed dependencies among these variables can produce interpretable visual semantic graphs, but continuous directed acyclic graph learning is limited by the cost of enforcing acyclicity. We present polyDAG, a polynomial acyclicity framework for efficient continuous causal discovery in visual semantic graphs. polyDAG replaces the matrix-exponential acyclicity constraint with a finite polynomial trace constraint and proves that the new constraint is zero exactly for acyclic graphs. We further derive a geometric-series implementation that avoids the explicit summation loop while preserving the same acyclicity condition. Experiments on synthetic Erdos-Renyi graphs and CelebA facial visual attributes show that polyDAG improves efficiency and structure recovery. Averaged over the revised synthetic protocol with d in {100, 200, 500}, polyDAG reduces mean structural Hamming distance from 318.4 to 285.4 and improves mean F1 score from 0.725 to 0.756. At 100 nodes, the geometric variant runs in 3.44 seconds compared with 5.16 seconds for the exponential baseline, corresponding to a 33.4 percent speedup. Code and data are publicly available at https://github.com/wenhaoz-fengcai/polyDAG.

Neural network (NN)-based nonlinear causal discovery methods recover DAG structure but leave each causal mechanism as a black box. Waxman et al. argued that extracting causal mechanisms from NN weights is ill-posed. We propose EML-CD, a framework that integrates the EML operator (capable of composing elementary functions from a single binary operator) into causal structure learning, with interpretable mechanism recovery as the primary objective. EML-CD represents each edge mechanism as a gated EML binary tree and automatically discovers closed-form causal equations. Analytical Jacobians can be directly computed from the output equations, enabling quantitative understanding of causal effects. On real data (Sachs protein signaling, d=11), EML-CD achieves SHD=11.2 +/- 0.4 (5-seed mean; baselines are single deterministic runs), on par with PC/GES within seed variance and below CAM, while attaching closed-form equations to each detected edge (precision 0.756, recall 0.365). In a controlled bivariate test with known mechanisms, EML-CD recovers 10 of 11 elementary function families faithfully (held-out shape correlation >= 0.96; only high-frequency sine is partial). On a symbolic synthetic benchmark, EML-CD attains a substantially lower and more stable held-out mechanism f-MSE than a fixed SINDy dictionary (mean 3.67 vs. 7644, the latter inflated by catastrophic extrapolation on one seed), although its structure recovery (SHD 14.0) only matches the dictionary and stays below specialized optimizers; on the Causal Chambers light-tunnel subset, a depth-2 model improves F1 over linear OLS-BIC (0.444 vs. 0.273).

Understanding potential selection in data is crucial for causal discovery; we argue that "selection" in common narratives takes two forms, which we term static and evolutionary selection, respectively. Static selection refers to a one-shot filtering process where observed data consist of a subset of the population of interest, as in survey volunteer bias. Evolutionary selection, in contrast, operates through repeated rounds of differential fitness in reproduction, where observed data constitute the latest generation shaped by a historical trajectory, as in immune adaptation, antibiotic resistance, and social norm emergence. Existing methods largely conflate these two forms and rely on an identical graphical model of selection. We show that this model is valid for static settings but fails to characterize data under evolution, yielding false discovery results. To address this, we introduce a new model that specifically characterizes evolutionary selection, and develop a sound and complete procedure for identifying such models from data across one or multiple environments or generations. Experimental results validate the method's ability to uncover the relevant mechanisms underlying evolution from data.

Textual event records, such as alarm logs, have become an increasingly common data source in engineering and manufacturing systems. Beyond identifying correlations or recurring patterns, engineers are often interested in understanding which types of events causally trigger or influence other events during system operation. Textual event descriptions may contain semantic clues about such causal relationships, and recent large language models (LLMs) provide a promising tool for extracting these signals. However, relying solely on LLM-encoded textual information is insufficient for accurate causal discovery, since semantic patterns do not directly reveal causal mechanisms and may confuse causation with correlation or frequent sequential patterns. To address these challenges, we propose \textbf{LMT}, a Bayesian causal discovery framework for engineering event data that jointly leverages textual descriptions and timestamps. Specifically, LMT first uses LLMs to extract semantic causal signals from event descriptions and constructs a prior distribution over causal graphs among event types or event clusters. It then incorporates temporal evidence through a Poisson-process-based likelihood, allowing the LLM-informed prior to be refined by timestamp-based statistical evidence. By integrating the textual and temporal information, LMT produces a causal graph that is both interpretable and data-supported. Simulation studies show that the proposed framework is effective across different settings and is especially advantageous in small-sample alarm-event scenarios.

Causal discovery from observational data remains challenging due to the fundamental limitations of purely statistical methods, such as statistical distinguishability within equivalence classes and sensitivity to finite sample sizes. While large language models (LLMs) offer a promising source of domain knowledge to complement statistical inference, existing LLM-augmented methods are vulnerable to LLM errors and incur high token costs. Moreover, reliance on a single data-centric algorithm can make results sensitive to algorithm-specific biases. To address these limitations, we propose CauTion, a framework that reliably integrates LLM domain knowledge into an ensemble of statistical causal discovery algorithms through consensus filtering and LLM reliability estimation. CauTion proceeds in three stages. First, an algorithm ensemble utilizes a consensus voting to resolve up to 96% of edges on which algorithms agree, achieving near-perfect accuracy on the filtered consensus edges. Second, a trust-calibrated arbitration mechanism estimates the relative reliability of the LLM and the algorithms via an annotation-free trust calibration procedure, which is then utilized to govern a trust-weighted voting process that restricts LLM arbitration exclusively to edges with unreliable algorithmic evidence. Third, a cycle repair step is applied to guarantee the final causal graph is validly acyclic. Experiments on six datasets demonstrate that CauTion consistently outperforms both data-centric and LLM-augmented baselines, with larger gains on larger graphs and strong robustness to LLM errors. Code is available at https://github.com/OpenCausaLab/CauTion.

Causal discovery with instantaneous effects in multivariate time series is challenging, as the instantaneous structure must be acyclic. Prior methods enforce this by either separating instantaneous and lagged estimation into multi-stage pipelines or imposing algebraic acyclicity constraints via complex augmented Lagrangian optimization, both of which incur high computational cost. In this work, we propose a different approach: we learn a differentiable permutation of variables using the Gumbel--Sinkhorn operator and triangularize the instantaneous coefficient matrix of a Structural Vector Autoregressive (SVAR) model in the learned order. This converts acyclicity from a hard constraint into a parameterization and keeps it valid throughout optimization. In doing so, our method enables unified, continuous optimization with gradient-based learning, leading to improved efficiency in time--series causal discovery. Across three real-world benchmarks, our method achieves the best overall performance compared with 12 baselines in both discovery accuracy and efficiency. On the large-scale benchmark, it further demonstrates strong scalability, achieving more than a 6x speedup over competing methods.

Temporal systems often exhibit non-stationary behaviour, such as seasonal climate variation or glucose fluctuations in patients with type-1 diabetes. One way to model non-stationarity is through discrete latent regimes, i.e., stationary segments of time. Such systems induce a Markov Switching Model (MSM), a class of Hidden Markov Models with autoregressive dependencies among latent regimes and observed variables. Identifying latent regimes is challenging in the presence of frequent regime switches and nonlinear and non-Gaussian dynamics, particularly when there are instantaneous effects between the variables, e.g., due to slow rates of measurements. In this work, we establish the identifiability of both latent regimes and regime-dependent causal structures under temporal regime dependencies, nonlinear lagged and instantaneous effects, and independent noise from the exponential family. Our identifiability theory subsumes non-temporal mixtures of causal models. Furthermore, we introduce FlowMSM, a regime detection framework that can be paired with any stationary causal discovery method to recover regime-dependent causal structures. Experiments on synthetic benchmarks and a financial economics dataset demonstrate the effectiveness of our approach to detect latent regimes and discover causal structures from non-stationary time series.

In graphical causal model, causal discovery aims to construct a causal graph based on numerical data and domain knowledge in plain text. However, the evaluation of causal discovery methods remains a challenge in the area as the progress of domain researches often makes benchmark causal graphs contain mis-aligned knowledge. This problem especially affects the evaluation of large language model (LLM) based causal discovery methods as they are sensitive to the new discoveries in the literature. This work is the first to systematically study the quality of benchmark causal graphs. Specifically, we design a pipeline that automatically retrieves relevant research papers from scientific databases, and prompts LLMs to check the consistency between the benchmark causal graphs and domain research papers. We evaluate 11 popular real-world benchmarks, for which our pipeline in total proceeds 38,081 domain papers. Our results show that popular benchmarks vary significantly in their consistency with domain research, with clear implications for causal discovery research.

Causal discovery aims to recover directed causal relations from observational and interventional data, providing a basis for mechanistic understanding and reliable decision-making. Causal discovery foundation models (CDFMs) seek to amortize this problem by mapping a dataset directly to a causal graph in a single forward pass, avoiding per-dataset testing, search, or optimization. However, existing CDFMs remain limited, often failing to consistently match strong classical methods, and we find that a key bottleneck is how causal pretraining tasks are constructed. Based on this observation, we propose TabCausal, a data-driven CDFM trained with broad causal pretraining over diverse graph priors, structural mechanisms, noise models, dimensions, sample sizes, and intervention regimes. A dynamic task construction strategy composes these causal environments into varied discovery tasks, enabling more transferable structural learning from observational and mixed-interventional data. On large-scale synthetic benchmarks, TabCausal achieves better macro-averaged performance than a diverse set of causal discovery baselines. To further bridge abstract synthetic generators and realistic causal reasoning scenarios, we introduce a protocol-guided and LLM-audited semantic causal environment benchmark, where domain-grounded SCMs generate interpretable observational and interventional datasets for out-of-distribution analysis. Across both synthetic and semantic environments, TabCausal demonstrates robust structure recovery, especially under interventional evidence, highlighting broad causal pretraining as a key ingredient for transferable amortized causal discovery.

Supervised Causal Learning (SCL) has shown promise in causal discovery by framing it as a supervised learning problem. However, it suffers from significant out-of-distribution generalization challenges. We reveal three limitations of previous SCL practices: a significant performance gap between synthetic benchmarks and real-world data, fragility to distribution shifts, and failure in compositional generalization, collectively questioning its real-world applicability. To address this, we propose Test-Time Training for Supervised Causal Learning (TTT-SCL), a novel framework that dynamically generates training sets explicitly aligned with any specific test instance. We demonstrate the correlation between TTT-SCL and score-based methods, and design an efficient module for generating training sets based on the classic scoring function. Experiments on synthetic benchmarks, pseudo-real and real-world datasets demonstrate that TTT-SCL significantly outperforms existing SCL and traditional causal discovery methods.

Causal analysis is a crucial task in many domains, including manufacturing, social science, and medicine. However, despite recent progress, the conceptual and methodological complexity of causal methods makes them largely inaccessible to domain experts. This gap prevents experts from leveraging these advances and hinders researchers who lack access to real-world data for validation. To bridge this divide, we introduce ORCA, a copilot for end-to-end causal analysis. ORCA orchestrates agents to understand the user's goals and guide them through the most appropriate causal analysis workflow, from fully automatic to highly user-guided execution. It features causal discovery, causal effect estimation, explainability and Root-Cause-Analysis (RCA). ORCA evaluates and compares performance, generates key metrics and diagrams, and generates insights through structured reports. We highlight its effectiveness across several real-world use-cases.