Causal Discovery

Multivariate time series in domains such as finance, climate science, and healthcare often exhibit long-term trends, seasonal patterns, and short-term fluctuations, complicating causal inference under non-stationarity and autocorrelation. Existing causal discovery methods typically operate on raw observations, making them vulnerable to spurious edges and misattributed temporal dependencies. We introduce a decomposition-based causal discovery framework that separates each time series into trend, seasonal, and residual components and performs component-specific causal analysis. Trend components are assessed using stationarity tests, seasonal components using kernel-based dependence measures, and residual components using constraint-based causal discovery. The resulting component-level graphs are integrated into a unified multi-scale causal structure. This approach isolates long- and short-range causal effects, reduces spurious associations, and improves interpretability. Across extensive synthetic benchmarks and real-world climate data, our framework more accurately recovers ground-truth causal structure than state-of-the-art baselines, particularly under strong non-stationarity and temporal autocorrelation.

To distinguish Markov equivalent graphs in causal discovery, it is necessary to restrict the structural causal model. Crucially, we need to be able to distinguish cause $X$ from effect $Y$ in bivariate models, that is, distinguish the two graphs $X \to Y$ and $Y \to X$. Location-scale noise models (LSNMs), in which the effect $Y$ is modeled based on the cause $X$ as $Y = f(X) + g(X)N$, form a flexible class of models that is general and identifiable in most cases. Estimating these models for arbitrary noise terms $N$, however, is challenging. Therefore, practical estimators are typically restricted to symmetric distributions, such as the normal distribution. As we showcase in this paper, when $N$ is a skewed random variable, which is likely in real-world domains, the reliability of these approaches decreases. To approach this limitation, we propose SkewD, a likelihood-based algorithm for bivariate causal discovery under LSNMs with skewed noise distributions. SkewD extends the usual normal-distribution framework to the skew-normal setting, enabling reliable inference under symmetric and skewed noise. For parameter estimation, we employ a combination of a heuristic search and an expectation conditional maximization algorithm. We evaluate SkewD on novel synthetically generated datasets with skewed noise as well as established benchmark datasets. Throughout our experiments, SkewD exhibits a strong performance and, in comparison to prior work, remains robust under high skewness.

Uncovering the causal mechanisms of complex real-world systems remains a significant challenge, as these systems often entail high data collection costs and involve unknown interventions. We introduce MetaCaDI, the first framework to cast the identification of unknown interventions as a meta-learning problem, explicitly leveraging a jointly learned causal graph. MetaCaDI is a Bayesian framework that learns a shared causal structure across multiple environments and is optimized to rapidly adapt to new, few-shot intervention target identification tasks. A key innovation is our model's analytical adaptation, which uses a closed-form solution to bypass expensive and potentially unstable gradient-based bilevel optimization. Extensive experiments on synthetic and complex gene expression data demonstrate that MetaCaDI significantly outperforms state-of-the-art methods. It excels at identifying intervention targets from as few as 3 samples - where existing methods collapse to random chance - while robustly recovering the shared causal graph, proving its effectiveness in data-scarce scenarios.

Granger causality characterizes directed predictive dependencies in multivariate time series, but recovering such dependencies becomes challenging in the presence of latent confounding. Cross-environment invariance provides a natural source of information in heterogeneous settings, yet invariance alone can be insufficient: when latent-to-observed mechanisms remain stable, hidden confounders can induce predictive dependencies that are just as invariant as genuine Granger-causal relations. We show that interventions provide an additional source of identifying information by inducing structured variation in observed mechanisms, while stable latent pathways need not exhibit the same cross-environment changes. In practice, however, neither the intervened environments nor the affected mechanisms are known. We propose GRACE, a framework for learning Granger causality under latent confounding from intervention-induced heterogeneity. GRACE decomposes multivariate dynamics into a shared Granger mechanism, sparse environment-specific deviations that capture edge-level interventions, and a latent component that accounts for confounding. Under a linear generative model, we show that GRACE can recover which environments intervene on a given edge when the edge is perturbed in at least one but fewer than half of the environments and the intervention effect is sufficiently large to survive sparsity shrinkage; the recovered intervention pattern then provides a certificate for the corresponding Granger causal edge. Experiments on synthetic and real-world time series demonstrate improved Granger causal structure recovery under latent confounding and unknown interventions.

We study the design of interventions for causal discovery in simple structural causal models whose causal graphs are directed mixed graphs (DMGs) that may contain directed cycles and bidirected edges representing latent confounding. In such case, observational conditional-independence (CI) information may not identify even the graph skeleton, while CI alone cannot generally detect a bidirected edge coexisting with a directed edge. To this end, we propose a stage-wise framework based on tailored separating systems. Separating-system interventions first recover descendant relations and strongly connected components (SCCs). The SCCs are then ordered by ancestry, and an SCC-Anc separating system recovers the directed subgraph. Given this subgraph, further systems use CI tests interpreted through $d$- or $σ$-separation to recover non-adjacent bidirected edges, whose endpoints share no directed edge, and do-see comparisons to recover those coexisting with exactly one directed edge. Under our assumptions, the framework recovers the directed subgraph and every bidirected edge except double-adjacent ones, whose endpoints are connected by a directed edge in each direction. We develop algorithms for unrestricted and $M$-bounded settings, with each experiment targeting at most $M$ variables in the latter. For recovering the directed subgraph and non-adjacent bidirected edges, our upper bounds on the number and maximum size of experiments match corresponding worst-case lower bounds up to logarithmic factors.

Causal discovery has shifted from data-centric methods to hybrid strategies that integrate semantic knowledge from experts or large language models (LLMs). Such external information is vital for identifying causal structures beyond the Markov Equivalence Class (MEC), which data alone cannot resolve. However, expert availability is often limited, and LLMs frequently misidentify causal directions in specialized domains. To overcome such shortcomings, we propose a tag-based approach that leverages semantically meaningful labels while deriving causal directionality directly from data. Using variable-level tag assignments from available sources (e.g., LLMs), our tags for DAGs method learns from identifiable data structures to extract higher-level causal relations. These are then used to orient undirected edges, enabling causal discovery to move beyond the MEC without reliance on fallible external knowledge.

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.

For two causal structures with the same set of visible variables, one is said to observationally dominate the other if the set of distributions over the visible variables realizable by the first contains the set of distributions over the visible variables realizable by the second. Knowing such dominance relations is useful for adjudicating between these structures given observational data. Here, we consider the problem of determining the partial order of equivalence classes of causal structures with latent variables relative to observational dominance. We provide a complete characterization of the dominance order in the case of three visible variables, and a partial characterization in the case of four visible variables. Our techniques also help to identify which observational equivalence classes have a set of realizable distributions that is characterized by nontrivial inequality constraints, analogous to Bell inequalities and instrumental inequalities. We find evidence that as one increases the number of visible variables, the equivalence classes satisfying nontrivial inequality constraints become ubiquitous. (Because such classes are the ones for which there can be a difference in the distributions that are quantumly and classically realizable, this implies that the potential for quantum-classical gaps is also ubiquitous.) Furthermore, we find evidence that constraint-based causal discovery algorithms that rely solely on conditional independence constraints have a significantly weaker distinguishing power among observational equivalence classes than algorithms that go beyond these (i.e., algorithms that also leverage nested Markov constraints and inequality constraints).

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.

The ability to understand causality significantly impacts the competence of large language models (LLMs) in output explanation and counterfactual reasoning, as causality reveals the underlying data distribution. However, the lack of a comprehensive benchmark currently limits the evaluation of LLMs' causal learning capabilities. To fill this gap, this paper develops CausalBN-Bench based on data from the causal research community, enabling comparative evaluations of LLMs against traditional causal learning algorithms. To provide a comprehensive investigation, we offer three tasks of varying difficulties, including correlation, causal skeleton, and causality identification. Evaluations of 19 leading LLMs reveal that, while closed-source LLMs show potential for simple causal relationships, they significantly lag behind traditional algorithms on larger-scale networks ($>50$ nodes). Specifically, LLMs struggle with collider structures but excel at chain structures, especially at long-chain causality analogous to Chain-of-Thought techniques. This supports the current prompt approaches while suggesting directions to enhance LLMs' causal reasoning capability. Furthermore, CausalBN-Bench incorporates background knowledge and training data into prompts to thoroughly unlock LLMs' text-comprehension ability during evaluation, whose findings indicate that, LLM understand causality through semantic associations with distinct entities, rather than directly from contextual information or numerical distributions.

Revealing hidden causal variables alongside the underlying causal mechanisms is essential to the development of science. Despite the progress in the past decades, existing practice in causal discovery (CD) heavily relies on high-quality measured variables, which are usually given by human experts. In fact, the lack of well-defined high-level variables behind unstructured data has been a longstanding roadblock to a broader real-world application of CD. This procedure can naturally benefit from an automated process that can suggest potential hidden variables in the system. Interestingly, Large language models (LLMs) are trained on massive observations of the world and have demonstrated great capability in processing unstructured data. To leverage the power of LLMs, we develop a new framework termed Causal representatiOn AssistanT (COAT) that incorporates the rich world knowledge of LLMs to propose useful measured variables for CD with respect to high-value target variables on their paired unstructured data. Instead of directly inferring causality with LLMs, COAT constructs feedback from intermediate CD results to LLMs to refine the proposed variables. Given the target variable and the paired unstructured data, we first develop COAT-MB that leverages the predictivity of the proposed variables to iteratively uncover the Markov Blanket of the target variable. Built upon COAT-MB, COAT-PAG further extends to uncover a more complete causal graph, i.e., Partial Ancestral Graph, by iterating over the target variables and actively seeking new high-level variables. Moreover, the reliable CD capabilities of COAT also extend the debiased causal inference to unstructured data by discovering an adjustment set. We establish theoretical guarantees for the CD results and verify their efficiency and reliability across realistic benchmarks and real-world case studies.

Causal discovery aims to learn causal structures up to certain symmetries. Diverse populations or changing environments give rise to heterogeneous data in the following sense: each population/environment is a ``source'' which idiosyncratically determines the forms of causal effects. From this perspective, the source is a latent common cause for every observed variable. While some methods for causal discovery can work around latent confounding in special cases, a global confounder poses a significant challenge. The only known ways to deal with latent global confounding involve making assumptions that limit structural equations and/or noise functions. We demonstrate that globally confounded causal structures can still be identified with arbitrary structural equations and noise functions, so long as the number of latent classes remains small relative to the size and sparsity of the underlying DAG. The approach relies on agglomerating variables into large-enough matrices of moments, whose ranks directly reveal graphical properties of the causal structure. We also provide a statistical test to test the rank of these matrices.