A new approach is introduced that estimates a global structure while accounting for local cluster-level effects, and presents a differentiable graph coupling mechanism that guarantees the union of the fixed- and random-effects graphs remains acyclic.
Abstract
Recent algorithmic advances have made directed acyclic graph (DAG) structure learning scalable for causal discovery. Yet, the currently available techniques assume a completely homogeneous population, precluding their application to clustered data where cluster-specific variations (e.g., patient-specific effects) are common. We address this issue by introducing a new approach that estimates a global structure while accounting for local cluster-level effects. The key idea is to extend the fixed- and random-effects framework of classical mixed models to the structure learning setting. Towards this end, we present a differentiable graph coupling mechanism that guarantees the union of the fixed- and random-effects graphs remains acyclic. Computationally, we provide a provably convergent first-order method and leverage efficient batched updates across clusters. Statistically, we establish identifiability of the model and show that our approach recovers the true structure asymptotically. In experiments on real and synthetic data, our proposal detects dependencies missed by alternative estimators, underscoring its value for structure learning in clustered settings.
Learning the causality between variables, known as DAG structure learning, is critical yet challenging due to issues such as insufficient data and noise. While prior knowledge can improve the learning process and refine the DAG structure, incorporating prior knowledge is not without pitfalls. In particular, we find that the gap between the imprecise prior knowledge and the exact weights modeled by existing methods may result in deviation in edge weights. Such deviation can subsequently cause significant inaccuracies when learning the DAG structure. This paper addresses this challenge by providing a theoretical analysis of the impact of deviation in edge weights during the optimization process of structure learning. We identify two special graph patterns that arise due to the deviation and show that their occurrence increases as the degree of deviation grows. Building on this analysis, we propose the P attern-G uided A daptive P rior (PGAP) framework. PGAP detects these patterns as structural signals during optimization and adaptively adjusts the structure learning process to counteract the identified weight deviation, thereby improving the integration of prior knowledge. Experiments verify the effectiveness and robustness of the proposed method.
Lyuzhou Chen, Yijia Sun, Yan Gao et al.· Neural Information Processin...· 0 citations
Causal discovery aims to recover the underlying directed acyclic graph (DAG) from observational data. Ordering-based methods offer a scalable perspective to global DAG search by estimating a topological order and assigning edges accordingly. However, existing approaches depend on repeated full-graph score evaluations and assume that local score minima correspond to causal sinks, an assumption that breaks down under statistical noise, model misspecification, or dense local dependencies. To overcome these limitations, we propose HiTOC, a Hierarchical Topological Ordering framework for Causal discovery that constructs the global causal structure via layer-wise integration of local orderings over induced subsets by Markov Blanket. At each iteration, nodes identified as sinks through local score-based rankings are peeled off to form hierarchical layers, avoiding global permutation search. HiTOC provides a clearer hierarchical structure via recursive modular inference and sink extraction, enabling interpretable layer-wise inference and localizing potential errors to small subgraphs. To ensure robustness under local inconsistencies, we further introduce a calibration mechanism with theoretical guarantees. Empirical results demonstrate that HiTOC achieves state-of-the-art performance in both accuracy and scalability, particularly in high-dimensional and structurally complex settings.
Haixiang Sun, Pengchao Tian, Zihan Zhou et al.· International Journal of Dat...· 0 citations
This work proposes SURE-Ridge, a non-iterative, closed-form estimator for equal variance linear Gaussian SEM, which achieves the lowest structural Hamming distance in the small-sample regime and the lowest run time across all sample sizes tested, compared with NOTEARS, DAGMA, and GBNSL baselines.
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.
S. Ahn, A. Leite, José L Costa et al.· 0 citations
We propose a unified Bayesian framework namely robust DAG-Cholesky horseshoe (R-DACH) for joint directed acyclic graph (DAG) structure learning and precision matrix estimation in the high-dimensional proportional asymptotic regime $p/n \to c \in (0,\infty)$, under the scale mixture of normal errors. The construction places a global-local horseshoe-type prior directly on the strictly lower-triangular entries of the modified Cholesky factor of the DAG-Markov precision matrix, so that sparsity in the Cholesky parameters induces a coherent parent-set selection consistent with a topological ordering of the variables. A per-observation inverse-gamma scale mixture yields automatic robustness to heavy-tailed and contaminated observations and admits Student-$t$, Laplace, and slash distributions as special cases. We design a partially-collapsed blocked Gibbs sampler that traverses the joint space of orderings, sparsity patterns and continuous parameters. Simulations across $(n,p)$ configurations with $p$ up to several hundreds confirm the theoretical rates and demonstrate substantial gains over graphical-horseshoe, DAG-Wishart, and PC-based competitors under contamination. An application to RNA-seq gene-expression data from \emph{The Cancer Genome Atlas} reveals biologically interpretable regulatory structure that competing methods fail to recover.
Samaneh Nazari, Mohammad Arashi, Abdolnasser Sadeghkhani· 0 citations
Many common data dependencies can be characterized by graphs: time series data are sequential (chain graph), images appear as pixels (lattice graph), areal data are defined by neighboring units (spatial adjacency graph), etc. Graph trend filtering seeks to smooth and predict such data. However, classical trend filtering only incorporates the graph for estimation of the trend, which limits its adaptivity, and is brittle in the presence of missing data. Further, it lacks uncertainty quantification and faces certain computing challenges. We address these limitations with a comprehensive Bayesian framework for (graph-) dependent data. Our approach leverages the graph at three critical junctures: 1) the trend, to enable smoothing, imputation, and prediction; 2) the local shrinkage, to enhance adaptivity and precision; and 3) the MCMC sampling algorithm, to deliver scalable posterior (predictive) inference via sparse and banded operations. For the proposed graph-dependent shrinkage priors, we study the local concentration and adaptivity properties and establish conditions for posterior propriety. Simulation studies demonstrate that, relative to state-of-the-art frequentist and Bayesian alternatives, this framework provides more accurate point estimates, more precise interval estimates, and highly competitive computing. We apply our methods for spatio-temporal modeling and forecasting of local area unemployment data for every county in the continental U.S. during the 2020 COVID-19 unemployment shock.