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.

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.

Hallucination remains a central failure mode of large language models, but existing benchmarks operationalize it inconsistently across tasks such as summarization, question answering, retrieval-augmented generation, and agentic interaction. This fragmentation makes it unclear whether a mitigation that works in one setting actually reduces hallucinations across contexts. Current hallucination benchmarks either require human annotation and fixed references that may eventually be memorized, or rely on naturalistic observations often recorded in settings that are difficult to reproduce or test systematically. To enable further research on the root causes of hallucination, we introduce HALLUWORLD, an extensible benchmark framework grounded in an explicit reference-world formulation: a model hallucinates when it produces an observable claim that is false with respect to this reference world. Building on this view, we construct a family of synthetic and semi-synthetic benchmark environments in which the reference world is fully specified, the model's observable view is controlled, and hallucination labels can be generated automatically by construction. HALLUWORLD spans multiple settings that are classically representative for AI, i.e., gridworlds, chess, and realistic terminal tasks. This enables controlled variation of key factors such as world complexity, observability, temporal change, and source-conflict policy, allowing us to disentangle hallucinations into more fine-grained error categories. We evaluate frontier and open-weight language models across these settings and find consistent patterns across domains: perceptual hallucination on directly observed information is near-solved for frontier models, while multi-step state tracking and causal forward simulation are still difficult for frontier models, and are not generally solved by extended thinking.

We propose a joint order-based scoring framework for causal structure learning of directed acyclic graph (DAG) models under heterogeneous data settings. We show that leveraging heterogeneity improves the accuracy of causal ordering estimation. In the most favorable case, the causal ordering is identifiable up to two permutations. Building on this framework, we propose an order-based Bayesian method for Gaussian DAG models and establish its theoretical properties in the high-dimensional regime. For posterior inference over the space of orderings, we introduce a random-to-random (R2R) proposal neighborhood for the Metropolis-Hastings algorithm, which is theoretically motivated and exhibits efficient mixing behavior. Simulation studies confirm the strong empirical performance of the proposed method, and an application to single-nucleus RNA sequencing data from major depressive disorder demonstrates practical utility.

We refer the readers to ( Peters et al., 2017) for more detailed graphical terminology. We base our proof mostly on ( Kirsch, 2019). The first statement follows directly from the first theorem in ( Haviland, 1936). Without loss of generality, we reorder the variables according to reversed topological ordering, i.e. a Follows directly from Lemma 1. Lemma 4. Recall condition 2) in Causal de Finetti states that 8 i, 8 n 2 N: X The first equality holds by well-defindedness. The fourth equality follow from well-definedness.

Wefindalarge range of behavior that can be precisely characterized by a new measure ofconfounding strength.

Understanding treatment effect heterogeneity is vital for scientific and policy research.

Next, we introduce two additional approximations we use to apply Eq. (S.9). An SCM matches a set of assignments to a causal graph. This implies that the error of the approximation Eq. (S.13) is mainly dominated by the gradients of g at hao, and the variance ofnao. Specifically, we use a positive differentiable measure of the statistical dependence, denoted by I. PIDA measures disentanglement of representations for models that are trained from unsupervised data. As a result, we have the following: Minimizing Eq. (S.21) leads topdo(a,o)(ˆφa0) approaching p(ˆφa0|a), which as we have just shown, leads top(ˆφa0|a) approaching pdo(a)(ˆφa0).