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Entropy
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Bayesian Networks and Causal Discovery
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Bayesian Networks and Causal Discovery

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: 15 January 2026 | Viewed by 2742

Special Issue Editors

Prof. Dr. Xiaoguang Gao

E-Mail Website
Guest Editor
School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710129, China
Interests: Bayesian network; reinforcement learning; causality; complex system modeling
Dr. Zidong Wang

E-Mail Website
Guest Editor Assistant
Department of Computer Science, City University of Hong Kong, Hong Kong, China
Interests: causal discovery; Bayesian networks; complex system modeling; multi-objective optimization

Special Issue Information

Dear Colleagues,

A bedrock topic in artificial intelligence is the discovery of the precise causal representations underlying data. Therefore, causal discovery is important for understanding data’s underlying mechanisms. As a probabilistic graphical model, a BN is a directed acyclic graph (DAG) in which each node represents a random variable and the directed edges between nodes represent the dependencies between variables. These relationships are further quantified by a set of conditional probability distributions. BNs have been applied to explore causality in various fields, such as fault detection, medical support, reliability analysis, and so on.

We hope that this Special Issue will become a forum for researchers in the field of Bayesian networks and causal discovery. Therefore, we are seeking unpublished original papers and comprehensive reviews focused on (but not limited to) the following research areas:

  • Bayesian network modeling, including structure learning, parameter learning, and inference algorithms.
  • Recent, popular continuous optimization algorithms in causal discovery, e.g., graph neural networks, reinforcement learning, etc.
  • The combination of Bayes and neural networks, e.g., Bayesian neural networks and deep Bayesian learning.
  • Novel causal models to represent causality.
  • The application of BNs and causal discovery, e.g., expert systems, reliability analysis, etc.
  • Causal discovery under confounding factors, e.g., noise, faithfulness, sufficiency, knowledge, and small datasets.

Prof. Dr. Xiaoguang Gao
Guest Editor

Dr. Zidong Wang
Guest Editor Assistant

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Bayesian network
  • directed acyclic graphs
  • causal discovery
  • structural equation model
  • structure learning
  • parameter learning
  • causal inference
  • neural networks and their applications

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Published Papers (3 papers)

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Research

30 pages, 3270 KB  
Article
Tree–Hillclimb Search: An Efficient and Interpretable Threat Assessment Method for Uncertain Battlefield Environments
by Zuoxin Zeng, Jinye Peng and Qi Feng
Entropy 2025, 27(9), 987; https://doi.org/10.3390/e27090987 - 21 Sep 2025
Viewed by 315
Abstract
In uncertain battlefield environments, rapid and accurate detection, identification of hostile targets, and assessment of threat levels are crucial for supporting effective decision-making. Despite offering the advantage of structural transparency, traditional analytical methods rely on expert knowledge to construct models and often fail [...] Read more.
In uncertain battlefield environments, rapid and accurate detection, identification of hostile targets, and assessment of threat levels are crucial for supporting effective decision-making. Despite offering the advantage of structural transparency, traditional analytical methods rely on expert knowledge to construct models and often fail to comprehensively capture the non-linear causal relationships among complex threat factors. In contrast, data-driven methods excel at uncovering patterns in data but suffer from limited interpretability due to their black-box nature. Owing to probabilistic graphical modeling capabilities, Bayesian networks possess unique advantages in threat assessment. However, existing models are either constrained by the limitation of expert experience or suffer from excessively high complexity due to structure learning algorithms, making it difficult to meet the stringent real-time requirements of uncertain battlefield environments. To address these issues, this paper proposes a new method, the Tree–Hillclimb Search method—an efficient and interpretable threat assessment method specifically designed for uncertain battlefield environments. The core of the method is a structure learning algorithm constrained by expert knowledge—the initial network structure constructed from expert knowledge serves as a constraint, enabling the discovery of hidden causal dependencies among variables through structure learning. The model is then refined under these expert knowledge constraints and can effectively balance accuracy and complexity. Sensitivity analysis further validates the consistency between the model structure and the influence degree of threat factors, providing a theoretical basis for formulating hierarchical threat assessment strategies under resource-constrained conditions, which can effectively optimize sensor resource allocation. The Tree–Hillclimb Search method features (1) enhanced interpretability; (2) high predictive accuracy; (3) high efficiency and real-time performance; (4) actual impact on battlefield decision-making; and (5) good generality and broad applicability. Full article
(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)
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34 pages, 2435 KB  
Article
Bridging Intuition and Data: A Unified Bayesian Framework for Optimizing Unmanned Aerial Vehicle Swarm Performance
by Ruiguo Zhong, Zidong Wang, Hao Wang, Yanghui Jin, Shuangxia Bai and Xiaoguang Gao
Entropy 2025, 27(9), 897; https://doi.org/10.3390/e27090897 - 25 Aug 2025
Viewed by 721
Abstract
The swift growth of the low-altitude economic ecosystem and Unmanned Aerial Vehicle (UAV) swarm applications across diverse sectors presents significant challenges for engineering managers in terms of effective performance evaluation and operational optimization. Traditional evaluation methods often struggle with the inherent complexities, dynamic [...] Read more.
The swift growth of the low-altitude economic ecosystem and Unmanned Aerial Vehicle (UAV) swarm applications across diverse sectors presents significant challenges for engineering managers in terms of effective performance evaluation and operational optimization. Traditional evaluation methods often struggle with the inherent complexities, dynamic nature, and multi-faceted performance criteria of UAV swarms. This study introduces a novel Bayesian Network (BN)-based multicriteria decision-making framework that systematically integrates expert intuition with real-time data. By employing variance decomposition, the framework establishes theoretically grounded, bidirectional mapping between expert-assigned weights and the network’s probabilistic parameters, creating a unified model of subjective expertise and objective data. Comprehensive validation demonstrates the framework’s efficacy in identifying critical performance drivers, including environmental awareness, communication ability, and a collaborative decision. Ultimately, our work provides engineering managers with a transparent and adaptive tool, offering actionable insights to inform resource allocation, guide technology adoption, and enhance the overall operational effectiveness of complex UAV swarm systems. Full article
(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)
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8 pages, 576 KB  
Article
Minimax Bayesian Neural Networks
by Junping Hong and Ercan Engin Kuruoglu
Entropy 2025, 27(4), 340; https://doi.org/10.3390/e27040340 - 25 Mar 2025
Cited by 1 | Viewed by 853
Abstract
Robustness is an important issue in deep learning, and Bayesian neural networks (BNNs) provide means of robustness analysis, while the minimax method is a conservative choice in the classical Bayesian field. Recently, researchers have applied the closed-loop idea to neural networks via the [...] Read more.
Robustness is an important issue in deep learning, and Bayesian neural networks (BNNs) provide means of robustness analysis, while the minimax method is a conservative choice in the classical Bayesian field. Recently, researchers have applied the closed-loop idea to neural networks via the minimax method and proposed the closed-loop neural networks. In this paper, we study more conservative BNNs with the minimax method, which formulates a two-player game between a deterministic neural network and a sampling stochastic neural network. From this perspective, we reveal the connection between the closed-loop neural and the BNNs. We test the models on some simple data sets and study their robustness under noise perturbation, etc. Full article
(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)
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