Causal Discovery

Real-world data often violates the equal-variance assumption (homoscedasticity), making it essential to account for heteroscedastic noise in causal discovery. In this work, we explore heteroscedastic symmetric noise models (HSNMs), where the effect $Y$ is modeled as $Y = f(X) + \sigma(X)N$, with $X$ as the cause and $N$ as independent noise following a symmetric distribution. We introduce a novel criterion for identifying HSNMs based on the skewness of the score (i.e., the gradient of the log density) of the data distribution. This criterion establishes a computationally tractable measurement that is zero in the causal direction but nonzero in the anticausal direction, enabling the causal direction discovery. We extend this skewness-based criterion to the multivariate setting and propose \texttt{SkewScore}, an algorithm that handles heteroscedastic noise without requiring the extraction of exogenous noise. We also conduct a case study on the robustness of \texttt{SkewScore} in a bivariate model with a latent confounder, providing theoretical insights into its performance. Empirical studies further validate the effectiveness of the proposed method.

We address the common yet often-overlooked selection bias in interventional studies, where subjects are selectively enrolled into experiments. For instance, participants in a drug trial are usually patients of the relevant disease; A/B tests on mobile applications target existing users only, and gene perturbation studies typically focus on specific cell types, such as cancer cells. Ignoring this bias leads to incorrect causal discovery results. Even when recognized, the existing paradigm for interventional causal discovery still fails to address it. This is because subtle differences in _when_ and _where_ interventions happen can lead to significantly different statistical patterns. We capture this dynamic by introducing a graphical model that explicitly accounts for both the observed world (where interventions are applied) and the counterfactual world (where selection occurs while interventions have not been applied). We characterize the Markov property of the model, and propose a provably sound algorithm to identify causal relations as well as selection mechanisms up to the equivalence class, from data with soft interventions and unknown targets. Through synthetic and real-world experiments, we demonstrate that our algorithm effectively identifies true causal relations despite the presence of selection bias.

Conventional image compression methods are inadequate for intelligent analysis, as they overemphasize pixel-level precision while neglecting semantic significance and the interaction among multiple tasks. This paper introduces a Taskonomy-Aware Multi-Task Compression framework comprising (1) inter-coherent task grouping, which organizes synergistic tasks into shared representations to improve multi-task accuracy and reduce encoding volume, and (2) a conditional entropy-based directed acyclic graph (DAG) that captures causal dependencies among grouped representations. By leveraging parent representations as contextual priors for child representations, the framework effectively utilizes cross-task information to improve entropy model accuracy. Experiments on diverse vision tasks, including Keypoint 2D, Depth Z-buffer, Semantic Segmentation, Surface Normal, Edge Texture, and Autoencoder, demonstrate significant bitrate-performance gains, validating the method’s capability to reduce system entropy uncertainty. These findings underscore the potential of leveraging representation disentanglement, synergy, and causal modeling to learn compact representations, which enable efficient multi-task compression in intelligent systems.

Causal discovery, or identifying causal relationships from observational data, is a notoriously challenging task, with numerous methods proposed to tackle it. Despite this, in-the-wild evaluation of these methods is still lacking, as works frequently rely on synthetic data evaluation and sparse real-world examples under critical theoretical assumptions. Real-world causal structures, however, are often complex, evolving over time, non-linear, and influenced by unobserved factors, making it hard to decide on a proper causal discovery strategy. To bridge this gap, we introduce CausalRivers, the largest in-the-wild causal discovery benchmarking kit for time-series data to date. CausalRivers features an extensive dataset on river discharge that covers the eastern German territory (666 measurement stations) and the state of Bavaria (494 measurement stations). It spans the years 2019 to 2023 with a 15-minute temporal resolution. Further, we provide additional data from a flood around the Elbe River, as an event with a pronounced distributional shift. Leveraging multiple sources of information and time-series meta-data, we constructed two distinct causal ground truth graphs (Bavaria and eastern Germany). These graphs can be sampled to generate thousands of subgraphs to benchmark causal discovery across diverse and challenging settings. To demonstrate the utility of CausalRivers, we evaluate several causal discovery approaches through a set of experiments to identify areas for improvement. CausalRivers has the potential to facilitate robust evaluations and comparisons of causal discovery methods. Besides this primary purpose, we also expect that this dataset will be relevant for connected areas of research, such as time-series forecasting and anomaly detection. Based on this, we hope to push benchmark-driven method development that fosters advanced techniques for causal discovery, as is the case for many other areas of machine learning.

In Model-Based Reinforcement Learning (MBRL), incorporating causal structures into dynamics models provides agents with a structured understanding of the environments, enabling efficient decision. Empowerment as an intrinsic motivation enhances the ability of agents to actively control their environments by maximizing the mutual information between future states and actions. We posit that empowerment coupled with causal understanding can improve controllability, while enhanced empowerment gain can further facilitate causal reasoning in MBRL. To improve learning efficiency and controllability, we propose a novel framework, Empowerment through Causal Learning (ECL), where an agent with the awareness of causal dynamics models achieves empowerment-driven exploration and optimizes its causal structure for task learning. Specifically, ECL operates by first training a causal dynamics model of the environment based on collected data. We then maximize empowerment under the causal structure for exploration, simultaneously using data gathered through exploration to update causal dynamics model to be more controllable than dense dynamics model without causal structure. In downstream task learning, an intrinsic curiosity reward is included to balance the causality, mitigating overfitting. Importantly, ECL is method-agnostic and is capable of integrating various causal discovery methods. We evaluate ECL combined with $3$ causal discovery methods across $6$ environments including pixel-based tasks, demonstrating its superior performance compared to other causal MBRL methods, in terms of causal discovery, sample efficiency, and asymptotic performance.

Targeted and uniform interventions to a system are crucial for unveiling causal relationships. While several methods have been developed to leverage interventional data for causal structure learning, their practical application in real-world scenarios often remains challenging. Recent benchmark studies have highlighted these difficulties, even when large numbers of single-variable intervention samples are available. In this work, we demonstrate, both theoretically and empirically, that such datasets contain a wealth of causal information that can be effectively extracted under realistic assumptions about the data distribution. More specifically, we introduce a novel variant of interventional faithfulness, which relies on comparisons between the marginal distributions of each variable across observational and interventional settings, and we introduce a score on causal orders. Under this assumption, we are able to prove strong theoretical guarantees on the optimum of our score that also hold for large-scale settings. To empirically verify our theory, we introduce Intersort, an algorithm designed to infer the causal order from datasets containing large numbers of single-variable interventions by approximately optimizing our score. Intersort outperforms baselines (GIES, DCDI, PC and EASE) on almost all simulated data settings replicating common benchmarks in the field. Our proposed novel approach to modeling interventional datasets thus offers a promising avenue for advancing causal inference, highlighting significant potential for further enhancements under realistic assumptions.

Recovering the underlying Directed Acyclic Graph (DAG) structures from observational data presents a formidable challenge, partly due to the combinatorial nature of the DAG-constrained optimization problem. Recently, researchers have identified gradient vanishing as one of the primary obstacles in differentiable DAG learning and have proposed several DAG constraints to mitigate this issue. By developing the necessary theory to establish a connection between analytic functions and DAG constraints, we demonstrate that analytic functions from the set $\\{f(x) = c_0 + \\sum_{i=1}^{\infty}c_ix^i | \\forall i > 0, c_i > 0; r = \\lim_{i\\rightarrow \\infty}c_{i}/c_{i+1} > 0\\}$ can be employed to formulate effective DAG constraints. Furthermore, we establish that this set of functions is closed under several functional operators, including differentiation, summation, and multiplication. Consequently, these operators can be leveraged to create novel DAG constraints based on existing ones. Using these properties, we design a series of DAG constraints and develop an efficient algorithm to evaluate them. Experiments in various settings demonstrate that our DAG constraints outperform previous state-of-the-art comparators. Our implementation is available at https://github.com/zzhang1987/AnalyticDAGLearning.

Causal representation learning (CRL) aims at recovering latent causal variables from high-dimensional observations to solve causal downstream tasks, such as predicting the effect of new interventions or more robust classification. A plethora of methods have been developed, each tackling carefully crafted problem settings that lead to different types of identifiability. These different settings are widely assumed to be important because they are often linked to different rungs of Pearl's causal hierarchy, even though this correspondence is not always exact. This work shows that instead of strictly conforming to this hierarchical mapping, *many causal representation learning approaches methodologically align their representations with inherent data symmetries.* Identification of causal variables is guided by invariance principles that are not necessarily causal. This result allows us to unify many existing approaches in a single method that can mix and match different assumptions, including non-causal ones, based on the invariance relevant to the problem at hand. It also significantly benefits applicability, which we demonstrate by improving treatment effect estimation on real-world high-dimensional ecological data. Overall, this paper clarifies the role of causal assumptions in the discovery of causal variables and shifts the focus to preserving data symmetries.

Diagnosing the root cause of an anomaly in a complex interconnected system is a pressing problem in today’s cloud services and industrial operations. We propose In-Distribution Interventions (IDI), a novel algorithm that predicts root cause as nodes that meet two criteria: 1) Anomaly: root cause nodes should take on anomalous values; 2) Fix: had the root cause nodes assumed usual values, the target node would not have been anomalous. Prior methods of assessing the fix condition rely on counterfactuals inferred from a Structural Causal Model (SCM) trained on historical data. But since anomalies are rare and fall outside the training distribution, the fitted SCMs yield unreliable counterfactual estimates. IDI overcomes this by relying on interventional estimates obtained by solely probing the fitted SCM at in-distribution inputs. We present a theoretical analysis comparing and bounding the errors in assessing the fix condition using interventional and counterfactual estimates. We then conduct experiments by systematically varying the SCM’s complexity to demonstrate the cases where IDI’s interventional approach outperforms the counterfactual approach and vice versa. Experiments on both synthetic and PetShop RCD benchmark datasets demonstrate that IDI consistently identifies true root causes more accurately and robustly than nine existing state-of-the-art RCD baselines. Code will be released at https://github.com/nlokeshiisc/IDI_release.

Identifying the root causes of anomalies in multivariate time series is challenging due to the complex dependencies among the series. In this paper, we propose a comprehensive approach called AERCA that inherently integrates Granger causal discovery with root cause analysis. By defining anomalies as interventions on the exogenous variables of time series, AERCA not only learns the Granger causality among time series but also explicitly models the distributions of exogenous variables under normal conditions. AERCA then identifies the root causes of anomalies by highlighting exogenous variables that significantly deviate from their normal states. Experiments on multiple synthetic and real-world datasets demonstrate that AERCA can accurately capture the causal relationships among time series and effectively identify the root causes of anomalies.

Identifying latent representations or causal structures is important for good generalization and downstream task performance. However, both fields developed rather independently. We observe that several structure and representation identifiability methods, particularly those that require multiple environments, rely on exchangeable non--i.i.d. (independent and identically distributed) data. To formalize this connection, we propose the Identifiable Exchangeable Mechanisms (IEM) framework to unify key representation and causal structure learning methods. IEM provides a unified probabilistic graphical model encompassing causal discovery, Independent Component Analysis, and Causal Representation Learning. With the help of the IEM model, we generalize the Causal de Finetti theorem of Guo et al., 2022 by relaxing the necessary conditions for causal structure identification in exchangeable data. We term these conditions cause and mechanism variability, and show how they imply a duality condition in identifiable representation learning, leading to new identifiability results.

Causal discovery aims to learn causal relationships between variables from targeted data, making it a fundamental task in machine learning. However, causal discovery algorithms often rely on unverifiable causal assumptions, which are usually difficult to satisfy in real-world data, thereby limiting the broad application of causal discovery in practical scenarios. Inspired by these considerations, this work extensively benchmarks the empirical performance of various mainstream causal discovery algorithms, which assume i.i.d. data, under eight model assumption violations. Our experimental results show that differentiable causal discovery methods exhibit robustness under the metrics of Structural Hamming Distance and Structural Intervention Distance of the inferred graphs in commonly used challenging scenarios, except for scale variation. We also provide the theoretical explanations for the performance of differentiable causal discovery methods. Finally, our work aims to comprehensively benchmark the performance of recent differentiable causal discovery methods under model assumption violations, and provide the standard for reasonable evaluation of causal discovery, as well as to further promote its application in real-world scenarios.

Large language models (LLMs) have shown remarkable capability in natural language tasks, yet debate persists on whether they truly comprehend deep structure (i.e., core semantics) or merely rely on surface structure (e.g., presentation format). Prior studies observe that LLMs' performance declines when intervening on surface structure, arguing their success relies on surface structure recognition. However, surface structure sensitivity does not prevent deep structure comprehension. Rigorously evaluating LLMs' capability requires analyzing both, yet deep structure is often overlooked. To this end, we assess LLMs' comprehension ability using causal mediation analysis, aiming to fully discover the capability of using both deep and surface structures. Specifically, we formulate the comprehension of deep structure as direct causal effect (DCE) and that of surface structure as indirect causal effect (ICE), respectively. To address the non-estimability of original DCE and ICE --- stemming from the infeasibility of isolating mutual influences of deep and surface structures, we develop the corresponding quantifiable surrogates, including approximated DCE (ADCE) and approximated ICE (AICE). We further apply the ADCE to evaluate a series of mainstream LLMs (and the one with random weights), showing that most of them exhibit deep structure comprehension ability, which grows along with the prediction accuracy. Comparing ADCE and AICE demonstrates closed-source LLMs (e.g., GPT) rely more on deep structure, while open-source LLMs (e.g., Llama) are more surface-sensitive, which decreases with model scale. Theoretically, ADCE is a bidirectional evaluation, which measures both the sufficiency and necessity of deep structure changes in causing output variations, thus offering a more comprehensive assessment than accuracy, a common evaluation in LLMs. Our work provides new insights into LLMs' deep structure comprehension and offers novel methods for LLMs evaluation. The code for our project is available at [ADCE Project](https://github.com/OpenCausaLab/ADCE).

Testing conditional independence (CI) has many important applications, such as Bayesian network learning and causal discovery. Although several approaches have been developed for learning CI structures for observed variables, those existing methods generally fail to work when the variables of interest can not be directly observed and only discretized values of those variables are available. For example, if $X_1$, $\tilde{X}_2$ and $X_3$ are the observed variables, where $\tilde{X}_2$ is a discretization of the latent variable $X_2$, applying the existing methods to the observations of $X_1$, $\tilde{X}_2$ and $X_3$ would lead to a false conclusion about the underlying CI of variables $X_1$, $X_2$ and $X_3$. Motivated by this, we propose a CI test specifically designed to accommodate the presence of discretization. To achieve this, a bridge equation and nodewise regression are used to recover the precision coefficients reflecting the conditional dependence of the latent continuous variables under the nonparanormal model. An appropriate test statistic has been proposed, and its asymptotic distribution under the null hypothesis of CI has been derived. Theoretical analysis, along with empirical validation on various datasets, rigorously demonstrates the effectiveness of our testing methods.

Causal discovery with latent variables is an important and challenging problem. To identify latent variables and infer their causal relations, most existing works rely on the assumption that latent variables have pure children. Considering that this assumption is potentially restrictive in practice and not strictly necessary in theory, in this paper, by introducing the concept of homologous surrogate, we eliminate the need for pure children in the context of causal discovery with latent variables. The homologous surrogate fundamentally differs from the pure child in the sense that the latter is characterized by having strictly restricted parents while the former allows for much more flexible parents. We formulate two assumptions involving homologous surrogates and develop theoretical results under each assumption. Under the weaker assumption, our theoretical results imply that we can determine each variable's ancestors, that is, partially recover the causal graph. The stronger assumption further enables us to determine each variable's parents exactly, that is, fully recover the causal graph. Building on these theoretical results, we derive an algorithm that fully leverages the properties of homologous surrogates for causal graph recovery. Also, we validate its efficacy through experiments. Our work broadens the applicability of causal discovery. Our code is available at: https://github.com/XiuchuanLi/ICLR2025-CDHS

Discovering a unique causal structure is difficult due to both inherent identifiability issues, and the consequences of finite data. As such, uncertainty over causal structures, such as those obtained from a Bayesian posterior, are often necessary for downstream tasks. Finding an accurate approximation to this posterior is challenging, due to the large number of possible causal graphs, as well as the difficulty in the subproblem of finding posteriors over the functional relationships of the causal edges. Recent works have used Bayesian meta learning to view the problem of posterior estimation as a supervised learning task. Yet, these methods are limited as they cannot reliably sample from the posterior over causal structures and fail to encode key properties of the posterior, such as correlation between edges and permutation equivariance with respect to nodes. To address these limitations, we propose a Bayesian meta learning model that allows for sampling causal structures from the posterior and encodes these key properties. We compare our meta-Bayesian causal discovery against existing Bayesian causal discovery methods, demonstrating the advantages of directly learning a posterior over causal structure.

Transformers have exhibited exceptional capabilities in sequence modelling tasks, leveraging self-attention and in-context learning. Critical to this success are induction heads, attention circuits that enable copying tokens based on their previous occurrences. In this work, we introduce a novel synthetic framework designed to enable the theoretical analysis of transformers’ ability to dynamically handle causal structures. Existing works rely on Markov Chains to study the formation of induction heads, revealing how transformers capture causal dependencies and learn transition probabilities in-context. However, they rely on a fixed causal structure that fails to capture the complexity of natural languages, where the relationship between tokens dynamically changes with context. To this end, our framework varies the causal structure through interleaved Markov chains with different lags while keeping the transition probabilities fixed. This setting unveils the formation of Selective Induction Heads, a new circuit that endows transformers with the ability to select the correct causal structure in-context. We empirically demonstrate that attention-only transformers learn this mechanism to predict the next token by identifying the correct lag and copying the corresponding token from the past. We provide a detailed construction of a 3-layer transformer to implement the selective induction head, and a theoretical analysis proving that this mechanism asymptotically converges to the maximum likelihood solution. Our findings advance the theoretical understanding of how transformers select causal structures, providing new insights into their functioning and interpretability.

Noisy linear structural causal models (SCMs) in the presence of confounding variables are known to be identifiable if all confounding and noise variables are non-Gaussian and unidentifiable if all are Gaussian. The identifiability when only some are Gaussian remains concealed. We show that, in the presence of Gaussian noise, a linear SCM is uniquely identifiable provided that \emph{(i)} the number of confounders is at most the number of the observed variables, \emph{(ii)} the confounders do not have a Gaussian component, and \emph{(iii)} the causal structure of the SCM is known. If the third condition is relaxed, the SCM becomes finitely identifiable; more specifically, it belongs to a set of at most $n!$ linear SCMS, where $n$ is the number of observed variables. The confounders in all of these $n!$ SCMs share the same joint probability distribution function (PDF), which we obtain analytically. For the case where both the noise and confounders are Gaussian, we provide further insight into the existing counter-example-based unidentifiability result and demonstrate that every SCM with confounders can be represented as an SCM without confounders but with the same joint PDF.

Synthetic datasets generated by structural causal models (SCMs) are commonly used for benchmarking causal structure learning algorithms. However, the variances and pairwise correlations in SCM data tend to increase along the causal ordering. Several popular algorithms exploit these artifacts, possibly leading to conclusions that do not generalize to real-world settings. Existing metrics like $\operatorname{Var}$-sortability and $\operatorname{R^2}$-sortability quantify these patterns, but they do not provide tools to remedy them. To address this, we propose internally-standardized structural causal models (iSCMs), a modification of SCMs that introduces a standardization operation at each variable during the generative process. By construction, iSCMs are not $\operatorname{Var}$-sortable. We also find empirical evidence that they are mostly not $\operatorname{R^2}$-sortable for commonly-used graph families. Moreover, contrary to the post-hoc standardization of data generated by standard SCMs, we prove that linear iSCMs are less identifiable from prior knowledge on the weights and do not collapse to deterministic relationships in large systems, which may make iSCMs a useful model in causal inference beyond the benchmarking problem studied here. Our code is publicly available at: https://github.com/werkaaa/iscm.

Understanding the dynamics of causal structures is crucial for uncovering the underlying processes in time series data. Previous approaches rely on static assumptions, where contemporaneous and time-lagged dependencies are assumed to have invariant topological structures. However, these models fail to capture the evolving causal relationship between variables when the underlying process exhibits such dynamics. To address this limitation, we propose DyCAST, a novel framework designed to learn dynamic causal structures in time series using Neural Ordinary Differential Equations (Neural ODEs). The key innovation lies in modeling the temporal dynamics of the contemporaneous structure, drawing inspiration from recent advances in Neural ODEs on constrained manifolds. We reformulate the task of learning causal structures at each time step as solving the solution trajectory of a Neural ODE on the directed acyclic graph (DAG) manifold. To accommodate high-dimensional causal structures, we extend DyCAST by learning the temporal dynamics of the hidden state for contemporaneous causal structure. Experiments on both synthetic and real-world datasets demonstrate that DyCAST achieves superior or comparable performance compared to existing causal discovery models.