--Learning the structure of Bayesian networks (BNs) from data is challenging, especially for datasets involving a large number of variables. The recently proposed divide-and-conquer (D&D) strategies present a promising approach for learning large BNs. However, they still face a main issue of unstable learning accuracy across subproblems. In this work, we introduce the idea of employing structure learning ensemble (SLE), which combines multiple BN structure learning algorithms, to consistently achieve high learning accuracy. We further propose an automatic approach called Auto-SLE for learning near-optimal SLEs, addressing the challenge of manually designing high-quality SLEs. The learned SLE is then integrated into a D&D method. Extensive experiments firmly show the superiority of our method over D&D methods with single BN structure learning algorithm in learning large BNs, achieving accuracy improvement usually by 30% 225% on datasets involving 10,000 variables. These results indicate the significant potential of employing (automatic learning of) SLEs for scalable BN structure learning. Learning the structure of Bayesian networks (BNs) [1] from data has attracted much research interest, due to its wide applications in machine learning, statistical modeling, and causal inference [2]-[4].

Bayesian Network (BN) structure learning traditionally centralizes data, raising privacy concerns when data is distributed across multiple entities. This research introduces Federated GES (FedGES), a novel Federated Learning approach tailored for BN structure learning in decentralized settings using the Greedy Equivalence Search (GES) algorithm. FedGES uniquely addresses privacy and security challenges by exchanging only evolving network structures, not parameters or data. It realizes collaborative model development, using structural fusion to combine the limited models generated by each client in successive iterations. A controlled structural fusion is also proposed to enhance client consensus when adding any edge.

Weaknesses: The paper emphasizes its focus on causal structure learning. In doing so it assumes "causal sufficiency", that is, it assumes that there are no latent confounders of the measured variables. Generally, there are many latent confounders of the measured variables in most domains. In the past 20 years, there has been substantial progress in developing graphical representations and algorithms for learning equivalence classes of causal networks from observational data. When causal sufficiency is assumed, the learning of DAG structure is generally called Bayesian network structure learning, not causal structural learning, as in the title of the paper. It would be helpful for the paper to more prominently highlight this assumption.

In this work, we propose a novel method for Bayesian Networks (BNs) structure elicitation that is based on the initialization of several LLMs with different experiences, independently querying them to create a structure of the BN, and further obtaining the final structure by majority voting. We compare the method with one alternative method on various widely and not widely known BNs of different sizes and study the scalability of both methods on them. We also propose an approach to check the contamination of BNs in LLM, which shows that some widely known BNs are inapplicable for testing the LLM usage for BNs structure elicitation. We also show that some BNs may be inapplicable for such experiments because their node names are indistinguishable. The experiments on the other BNs show that our method performs better than the existing method with one of the three studied LLMs; however, the performance of both methods significantly decreases with the increase in BN size.

Dynamic Bayesian Networks (DBNs), renowned for their interpretability, have become increasingly vital in representing complex stochastic processes in various domains such as gene expression analysis, healthcare, and traffic prediction. Structure learning of DBNs from data is challenging, particularly for datasets with thousands of variables. Most current algorithms for DBN structure learning are adaptations from those used in static Bayesian Networks (BNs), and are typically focused on small-scale problems. In order to solve large-scale problems while taking full advantage of existing algorithms, this paper introduces a novel divide-and-conquer strategy, originally developed for static BNs, and adapts it for large-scale DBN structure learning. In this work, we specifically concentrate on 2 Time-sliced Bayesian Networks (2-TBNs), a special class of DBNs. Furthermore, we leverage the prior knowledge of 2-TBNs to enhance the performance of the strategy we introduce. Our approach significantly improves the scalability and accuracy of 2-TBN structure learning. Experimental results demonstrate the effectiveness of our method, showing substantial improvements over existing algorithms in both computational efficiency and structure learning accuracy. On problem instances with more than 1,000 variables, our approach improves two accuracy metrics by 74.45% and 110.94% on average , respectively, while reducing runtime by 93.65% on average.

Identifying functional connectivity biomarkers of major depressive disorder (MDD) patients is essential to advance understanding of the disorder mechanisms and early intervention. However, due to the small sample size and the high dimension of available neuroimaging data, the performance of existing methods is often limited. Multi-site data could enhance the statistical power and sample size, while they are often subject to inter-site heterogeneity and data-sharing policies. In this paper, we propose a federated joint estimator, NOTEARS-PFL, for simultaneous learning of multiple Bayesian networks (BNs) with continuous optimization, to identify disease-induced alterations in MDD patients. We incorporate information shared between sites and site-specific information into the proposed federated learning framework to learn personalized BN structures by introducing the group fused lasso penalty. We develop the alternating direction method of multipliers, where in the local update step, the neuroimaging data is processed at each local site. Then the learned network structures are transmitted to the center for the global update. In particular, we derive a closed-form expression for the local update step and use the iterative proximal projection method to deal with the group fused lasso penalty in the global update step. We evaluate the performance of the proposed method on both synthetic and real-world multi-site rs-fMRI datasets. The results suggest that the proposed NOTEARS-PFL yields superior effectiveness and accuracy than the comparable methods.

Traditionally, Bayesian network structure learning is often carried out at a central site, in which all data is gathered. However, in practice, data may be distributed across different parties (e.g., companies, devices) who intend to collectively learn a Bayesian network, but are not willing to disclose information related to their data owing to privacy or security concerns. In this work, we present a cross-silo federated learning approach to estimate the structure of Bayesian network from data that is horizontally partitioned across different parties. We develop a distributed structure learning method based on continuous optimization, using the alternating direction method of multipliers (ADMM), such that only the model parameters have to be exchanged during the optimization process. We demonstrate the flexibility of our approach by adopting it for both linear and nonlinear cases. Experimental results on synthetic and real datasets show that it achieves an improved performance over the other methods, especially when there is a relatively large number of clients and each has a limited sample size.

This paper addresses the problem of learning a sparse structure Bayesian network from high-dimensional discrete data. Compared to continuous Bayesian networks, learning a discrete Bayesian network is a challenging problem due to the large parameter space. Although many approaches have been developed for learning continuous Bayesian networks, few approaches have been proposed for the discrete ones. In this paper, we address learning Bayesian networks as an optimization problem and propose a score function that satisfies the sparsity and the DAG property simultaneously. Besides, we implement a block-wised stochastic coordinate descent algorithm to optimize the score function. Specifically, we use a variance reducing method in our optimization algorithm to make the algorithm work efficiently in high-dimensional data. The proposed approach is applied to synthetic data from well-known benchmark networks. The quality, scalability, and robustness of the constructed network are measured. Compared to some competitive approaches, the results reveal that our algorithm outperforms the others in evaluation metrics.