Causal Discovery

Uncovering cause-effect relationships from observational time series is fundamental to understanding complex systems. While many methods infer static causal graphs, real-world systems often exhibit *dynamic causality*—where relationships evolve over time. Accurately capturing these temporal dynamics requires time-resolved causal graphs. We propose UnCLe, a novel deep learning method for scalable dynamic causal discovery. UnCLe employs a pair of Uncoupler and Recoupler networks to disentangle input time series into semantic representations and learns inter-variable dependencies via auto-regressive Dependency Matrices. It estimates dynamic causal influences by analyzing datapoint-wise prediction errors induced by temporal perturbations. Extensive experiments demonstrate that UnCLe not only outperforms state-of-the-art baselines on static causal discovery benchmarks but, more importantly, exhibits a unique capability to accurately capture and represent evolving temporal causality in both synthetic and real-world dynamic systems (e.g., human motion). UnCLe offers a promising approach for revealing the underlying, time-varying mechanisms of complex phenomena.

Capturing the interactions between entities in a structured way plays a central role in world models that flexibly adapt to changes in the environment. Recent works motivate the benefits of models that explicitly represent the structure of interactions and formulate the problem as discovering local causal structures. In this work, we demonstrate that reliably capturing these relationships in complex settings remains challenging. To remedy this shortcoming, we postulate that sparsity is a critical ingredient for the discovery of such local structures. To this end we present the SPARse TrANsformer World model (SPARTAN), a Transformer-based world model that learns context-dependent interaction structures between entities in a scene. By applying sparsity regularisation on the attention patterns between object-factored tokens, SPARTAN learns sparse, context-dependent interaction graphs that accurately predict future object states. We further extend our model to adapt to sparse interventions with unknown targets on the dynamics of the environment. This results in a highly interpretable world model that can efficiently adapt to changes. Empirically, we evaluate SPARTAN against the current state-of-the-art in object-centric world models on observation-based environments and demonstrate that our model can learn local causal graphs that accurately reflects the underlying interactions between objects and achieve significantly improved few-shot adaptation to dynamics changes as well as robustness against distractors.

Distinguishing cause and effect from bivariate observational data is a foundational problem in many disciplines, but challenging without additional assumptions. Additive noise models (ANMs) are widely used to enable sample-efficient bivariate causal discovery. However, conventional ANM-based methods fail when unobserved mediators corrupt the causal relationship between variables. This paper makes three key contributions: first, we rigorously characterize why standard ANM approaches break down in the presence of unmeasured mediators. Second, we demonstrate that prior solutions for hidden mediation are brittle in finite sample settings, limiting their practical utility. To address these gaps, we propose Bivariate Denoising Diffusion (BiDD) for causal discovery, a method designed to handle latent noise introduced by unmeasured mediators. Unlike prior methods that infer directionality through mean squared error loss comparisons, our approach introduces a novel independence test statistic: during the noising and denoising processes for each variable, we condition on the other variable as input and evaluate the independence of the predicted noise relative to this input. We prove asymptotic consistency of BiDD under the ANM, and conjecture that it performs well under hidden mediation. Experiments on synthetic and real-world data demonstrate consistent performance, outperforming existing methods in mediator-corrupted settings while maintaining strong performance in mediator-free settings.

We study the problem of causal structure learning from a combination of observational and interventional data generated by a linear non-Gaussian structural equation model that might contain cycles. Recent results show that using mere observational data identifies the causal graph only up to a permutation-equivalence class. We obtain a combinatorial characterization of this class by showing that each equivalence class corresponds to a perfect matching in a bipartite graph. This bipartite representation allows us to analyze how interventions modify or constrain the matchings. Specifically, we show that each atomic intervention reveals one edge of the true matching and eliminates all incompatible causal graphs. Consequently, we formalize the optimal experiment design task as an adaptive stochastic optimization problem over the set of equivalence classes with a natural reward function that quantifies how many graphs are eliminated from the equivalence class by an intervention. We show that this reward function is adaptive submodular and provide a greedy policy with a provable near-optimal performance guarantee. A key technical challenge is to efficiently estimate the reward function without having to explicitly enumerate all the graphs in the equivalence class. We propose a sampling-based estimator using random matchings and analyze its bias and concentration behavior. Our simulation results show that performing a small number of interventions guided by our stochastic optimization framework recovers the true underlying causal structure.

Uncovering cause-and-effect mechanisms from data is fundamental to scientific progress. While large language models (LLMs) show promise for enhancing causal discovery (CD) from unstructured data, their application to the increasingly prevalent multimodal setting remains a critical challenge. Even with the advent of multimodal LLMs (MLLMs), their efficacy in multimodal CD is hindered by two primary limitations: (1) difficulty in exploring intra- and inter-modal interactions for comprehensive causal variable identification; and (2) insufficiency to handle structural ambiguities with purely observational data. To address these challenges, we propose MLLM-CD, a novel framework for multimodal causal discovery from unstructured data. It consists of three key components: (1) a novel contrastive factor discovery module to identify genuine multimodal factors based on the interactions explored from contrastive sample pairs; (2) a statistical causal structure discovery module to infer causal relationships among discovered factors; and (3) an iterative multimodal counterfactual reasoning module to refine the discovery outcomes iteratively by incorporating the world knowledge and reasoning capabilities of MLLMs. Extensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed MLLM-CD in revealing genuine factors and causal relationships among them from multimodal unstructured data. The implementation code and data are available at https://github.com/JinLi-i/MLLM-CD.

The do-calculus is a sound and complete tool for identifying causal effects in acyclic directed mixed graphs (ADMGs) induced by structural causal models (SCMs). However, in many real-world applications, especially in high-dimensional settings, constructing a fully specified ADMG is often infeasible. This limitation has led to growing interest in partially specified causal representations, particularly through cluster-directed mixed graphs (C-DMGs), which group variables into clusters and offer a more abstract yet practical view of causal dependencies. While these representations can include cycles, recent work has shown that the do-calculus remains sound and complete for identifying macro-level causal effects in C-DMGs over ADMGs under the assumption that all clusters sizes are greater than 1. Nevertheless, real-world systems often exhibit cyclic causal dynamics at the structural level. To account for this, input-output structural causal models (ioSCMs) have been introduced as a generalization of SCMs that allow for cycles. ioSCMs induce another type of graph structure known as a directed mixed graph (DMG). Analogous to the ADMG setting, one can define C-DMGs over DMGs as high-level representations of causal relations among clusters of variables. In this paper, we prove that, unlike in the ADMG setting, the do-calculus is unconditionally sound and complete for identifying macro causal effects in C-DMGs over DMGs. Furthermore, we show that the graphical criteria for non-identifiability of macro causal effects previously established C-DMGs over ADMGs naturally extends to a subset of C-DMGs over DMGs.

Existing methods for differentiable structure learning in discrete data typically assume that the data are generated from specific structural equation models. However, these assumptions may not align with the true data-generating process, which limits the general applicability of such methods. Furthermore, current approaches often ignore the complex dependence structure inherent in discrete data and consider only linear effects. We propose a differentiable structure learning framework that is capable of capturing arbitrary dependencies among discrete variables. We show that although general discrete models are unidentifiable from purely observational data, it is possible to characterize the complete set of compatible parameters and structures. Additionally, we establish identifiability up to the Markov equivalence class (MEC) under mild assumptions. We formulate the learning problem as a single differentiable optimization task in the most general form, thereby avoiding the unrealistic simplifications adopted by previous methods. Empirical results demonstrate that our approach effectively captures complex relationships in discrete data.

Greedy Equivalence Search (GES) is a classic score-based algorithm for causal discovery from observational data. In the sample limit, it recovers the Markov equivalence class of graphs that describe the data. Still, it faces two challenges in practice: computational cost and finite-sample accuracy. In this paper, we develop Less Greedy Equivalence Search (LGES), a variant of GES that retains its theoretical guarantees while partially addressing these limitations. LGES modifies the greedy step; rather than always applying the highest-scoring insertion, it avoids edge insertions between variables for which the score implies some conditional independence. This more targeted search yields up to a $10$-fold speed-up and a substantial reduction in structural error relative to GES. Moreover, LGES can guide the search using prior knowledge, and can correct this knowledge when contradicted by data. Finally, LGES can use interventional data to refine the learned observational equivalence class. We prove that LGES recovers the true equivalence class in the sample limit, even with misspecified knowledge. Experiments demonstrate that LGES outperforms GES and other baselines in speed, accuracy, and robustness to misspecified knowledge. Our code is available at https://github.com/CausalAILab/lges}{https://github.com/CausalAILab/lges.

Graph classification has long assumed a closed-world setting, limiting its applicability to real-world scenarios where new categories often emerge. To address this limitation, we introduce Graph-Level Novel Category Discovery (GLNCD), a new task aimed at identifying unseen graph categories without supervision from novel classes. We first adapt classical Novel Category Discovery (NCD) methods for images to the graph domain and evaluate these baseline methods on four diverse graph datasets curated for the GLNCD task. Our analysis reveals that these methods suffer a notable performance degradation compared to their image-based counterparts, due to two key challenges: (1) insufficient utilization of structural information in graph self-supervised learning (SSL), and (2) ineffective pseudo-labeling strategies based on ranking statistics (RS) that neglect graph structure. To alleviate these issues, we propose ProtoFGW-NCD, a framework consisting of two core components: ProtoFGW-CL, a novel graph SSL framework, and FGW-RS, a structure-aware pseudo-labeling method. Both components employ a differentiable Fused Gromov-Wasserstein (FGW) distance to effectively compare graphs by incorporating structural information. These components are built upon learnable prototype graphs, which enable efficient, parallel FGW-based graph comparisons and capture representative patterns within graph datasets. Experiments on four GLNCD benchmark datasets demonstrate the effectiveness of ProtoFGW-NCD.

Causal reasoning and discovery, two fundamental tasks of causal analysis, often face challenges in applications due to the complexity, noisiness, and high-dimensionality of real-world data. Despite recent progress in identifying latent causal structures using causal representation learning (CRL), what makes learned representations useful for causal downstream tasks and how to evaluate them are still not well understood. In this paper, we reinterpret CRL using a measurement model framework, where the learned representations are viewed as proxy measurements of the latent causal variables. Our approach clarifies the conditions under which learned representations support downstream causal reasoning and provides a principled basis for quantitatively assessing the quality of representations using a new Test-based Measurement EXclusivity (T-MEX) score. We validate T-MEX across diverse causal inference scenarios, including numerical simulations and real-world ecological video analysis, demonstrating that the proposed framework and corresponding score effectively assess the identification of learned representations and their usefulness for causal downstream tasks.

Causal discovery for dynamical systems poses a major challenge in fields where active interventions are infeasible. Most methods used to investigate these systems and their associated benchmarks are tailored to deterministic, low-dimensional and weakly nonlinear time-series data. To address these limitations, we present *CausalDynamics*, a large-scale benchmark and extensible data generation framework to advance the structural discovery of dynamical causal models. Our benchmark consists of true causal graphs derived from thousands of both linearly and nonlinearly coupled ordinary and stochastic differential equations as well as two idealized climate models. We perform a comprehensive evaluation of state-of-the-art causal discovery algorithms for graph reconstruction on systems with noisy, confounded, and lagged dynamics. *CausalDynamics* consists of a plug-and-play, build-your-own coupling workflow that enables the construction of a hierarchy of physical systems. We anticipate that our framework will facilitate the development of robust causal discovery algorithms that are broadly applicable across domains while addressing their unique challenges. We provide a user-friendly implementation and documentation on https://kausable.github.io/CausalDynamics.

Estimating causal effects from nonexperimental data is a fundamental problem in many fields of science. A key component of this task is selecting an appropriate set of covariates for confounding adjustment to avoid bias. Most existing methods for covariate selection often assume the absence of latent variables and rely on learning the global causal structure among variables. However, identifying the global structure can be unnecessary and inefficient, especially when our primary interest lies in estimating the effect of a treatment variable on an outcome variable. To address this limitation, we propose a novel local learning approach for covariate selection in nonparametric causal effect estimation, which accounts for the presence of latent variables. Our approach leverages testable independence and dependence relationships among observed variables to identify a valid adjustment set for a target causal relationship, ensuring both soundness and completeness under standard assumptions. We validate the effectiveness of our algorithm through extensive experiments on both synthetic and real-world data.

Real-world datasets are often a combination of unobserved subpopulations that follow distinct causal generating processes. In an observational study, for example, participants may fall into unknown groups that either (a) respond effectively to a drug, or (b) show no response due to drug resistance. Not accounting for such heterogeneity then risks biased estimates of drug effectiveness. In this work, we formulate this setting through a causal mixture model, in which the data-generating process of each variable depends on latent group membership (a or b). Specifically, we model each variable as a mixture of structural causal equation models, where latent categorical (mixing) variables index assignment to subpopulations. Unlike prior work, the approach allows for multiple independent mixing variables, each affecting distinct sets of observed variables. To infer both the graph, mixing variables, and assignments jointly, we integrate mixture modeling into score-based causal discovery; show theoretically that the resulting scoring criterion is consistent; and demonstrate empirically that the resulting causal discovery approach discovers the causal model in synthetic and real-world evaluations.

Language model activations entangle concepts that mediate their behavior, making it difficult to interpret these factors, which has implications for generalizability and robustness. We introduce an approach for disentangling these concepts without supervision. Existing methods for concept discovery often rely on external labels, contrastive prompts, or known causal structures, which limits their scalability and biases them toward predefined, easily annotatable features. In contrast, we propose a new unsupervised algorithm that identifies causal differentiating concepts—interpretable latent directions in LM activations that must be changed to elicit a different model behavior. These concepts are discovered using a constrained contrastive learning objective, guided by the insight that eliciting a target behavior requires only sparse changes to the underlying concepts. We formalize this notion and show that, under a particular assumption about the sparsity of these causal differentiating concepts, our method learns disentangled representations that align with human-interpretable factors influencing LM decisions. We empirically show the ability of our method to recover ground-truth causal factors in synthetic and semi-synthetic settings. Additionally, we illustrate the utility of our method through a case study on refusal behavior in language models. Our approach offers a scalable and interpretable lens into the internal workings of LMs, providing a principled foundation for interpreting language model behavior.

Causal discovery methods are powerful tools for uncovering the structure of relationships among variables, yet they face significant challenges in scalability and interpretability, especially in high-dimensional settings. In many domains, researchers are not only interested in causal links between individual variables, but also in relationships among sets or clusters of variables. Learning causal structure at the cluster level can both reveal higher-order relationships of interest and improve scalability. In this work, we introduce an approach for causal discovery over clusters in Markov causal systems. We propose a new graphical model that encodes knowledge of relationships between user-defined clusters while fully representing independencies and dependencies over clusters, faithful to a given distribution. We then define and characterize a graphical equivalence class of these models that share cluster-level independence information. Lastly, we present a sound and complete algorithm for causal discovery to represent learnable causal relationships between clusters of variables.

Causal discovery in the form of a directed acyclic graph (DAG) for dynamic time series data has been widely studied in various applications. Much of the existing work has focused on observational, offline, and/or stationary settings. In this work, we propose a dynamic DAG discovery algorithm, Meta-D$^2$AG, based on online meta-learning. Meta-D$^2$AG is designed to learn dynamic DAG structures from potentially nonlinear and non-stationary times series datasets, accounting for changes in both parameters and graph structures. Notably, Meta-D$^2$AG explicitly treats data collected at different time points with distribution shifts as distinct domains, which is assumed to occur as a result of external interventions. Moreover, Meta-D$^2$AG contains a new online meta-learning framework to take advantage of the temporal transition among existing domains such that it can quickly adapt to new domains with few measurements. A first-order optimization approach is utilized to efficiently solve the meta-learning framework, and theoretical analysis establishes the identifiability conditions and the convergence of the learning process. We demonstrate the promising performance of our method through better accuracy and sample efficiency on benchmark datasets against state-of-the-art baselines.

Ordering-based approaches to causal discovery identify topological orders of causal graphs, providing scalable alternatives to combinatorial search methods. Under the Additive Noise Models (ANMs) assumption, recent causal ordering methods based on score matching require an accurate estimation of the Hessian diagonal of the log-densities. However, previous approaches mainly use Stein gradient estimators, which are computationally expensive and memory-intensive. Although DiffAN addresses these limitations by substituting kernel-based estimates with diffusion models, it remains numerically unstable due to the second-order derivatives of score models. To alleviate these problems, we propose Score-informed Neural Operator (SciNO), a probabilistic generative model in smooth function spaces designed to stably approximate the Hessian diagonal and to preserve structural information during the score modeling. Empirical results show that SciNO reduces order divergence by 42.7% on synthetic graphs and by 31.5% in real-world datasets on average compared to DiffAN, while maintaining memory efficiency and scalability. Furthermore, we propose a probabilistic control algorithm for causal reasoning with autoregressive models that integrates SciNO's probability estimates with autoregressive model priors, enabling reliable data-driven causal ordering informed by semantic information. Consequently, the proposed method enhances causal reasoning abilities of LLMs without additional fine-tuning or prompt engineering.

In this paper, we explore the universal properties underlying causal inference by formulating it in terms of a topos. More concretely, we introduce topos causal models (TCMs), a strict generalization of the popular structural causal models (SCMs). A topos category has several properties that make it attractive: a general theory for how to combine local functions that define ``independent causal mechanisms" into a consistent global function building on the theory of sheaves in a topos; a generic way to define causal interventions using a subobject classifier in a topos category; and finally, an internal logical language for causal and counterfactual reasoning that emerges from the topos itself. A striking characteristic of subobject classifiers is that they induce an intuitionistic logic, whose semantics is based on the partially ordered lattice of subobjects. We show that the underlying subobject classifier for causal inference is not Boolean in general, but forms a Heyting algebra. We define the internal Mitchell-B\'enabou language, a typed local set theory, associated with causal models, and its associated Kripke-Joyal intuitionistic semantics. We prove a universal property of TCM, namely that any causal functor mapping decomposable structure to probabilistic semantics factors uniquely through a TCM representation.

In this paper, we focus on Novel Class Discovery for Point Cloud Segmentation (3D-NCD), aiming to learn a model that can segment unlabeled (novel) 3D classes using only the supervision from labeled (base) 3D classes. The key to this task is to setup the exact correlations between the point representations and their base class labels, as well as the representation correlations between the points from base and novel classes. A coarse or statistical correlation learning may lead to the confusion in novel class inference. lf we impose a causal relationship as a strong correlated constraint upon the learning process, the essential point cloud representations that accurately correspond to the classes should be uncovered. To this end, we introduce a structural causal model (SCM) to re-formalize the 3D-NCD problem and propose a new method, i.e., Joint Learning of Causal Representation and Reasoning. Specifically, we first analyze hidden confounders in the base class representations and the causal relationships between the base and novel classes through SCM. We devise a causal representation prototype that eliminates confounders to capture the causal representations of base classes. A graph structure is then used to model the causal relationships between the base classes' causal representation prototypes and the novel class prototypes, enabling causal reasoning from base to novel classes. Extensive experiments and visualization results on 3D and 2D NCD semantic segmentation demonstrate the superiorities of our method.

Conditional independence (CI) testing is a fundamental yet challenging task in modern statistics and machine learning. One pivotal class of methods for assessing conditional independence encompasses kernel-based approaches, known for assessing CI by detecting general conditional dependence without imposing strict assumptions on relationships or data distributions. As with any method utilizing kernels, selecting appropriate kernels is crucial for precise identification. However, it remains underexplored in kernel-based CI methods, where the kernels are often determined manually or heuristically. In this paper, we analyze and propose a kernel parameter selection approach for the kernel-based conditional independence test (KCI). The kernel parameters are selected based on the ratio of the statistic to the asymptotic variance, which approximates the test power for the given parameters at large sample sizes. The search procedure is grid-based, allowing for parallelization with manageable additional computation time. We theoretically demonstrate the consistency of the proposed criterion and conduct extensive experiments on both synthetic and real data to show the effectiveness of our method.