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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: closed (15 January 2026) | Viewed by 5146

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 Therefore, we are seeking unpublished original papers and comprehensive reviews focused on (but not limited to) the following research areas: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 250 words) can be sent to the Editorial Office for assessment.

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 (6 papers)

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Research

26 pages, 6232 KB  
Article
MFE-YOLO: A Multi-Scale Feature Enhanced Network for PCB Defect Detection with Cross-Group Attention and FIoU Loss
by Ruohai Di, Hao Fan, Hanxiao Feng, Zhigang Lv, Lei Shu, Rui Xie and Ruoyu Qian
Entropy 2026, 28(2), 174; https://doi.org/10.3390/e28020174 - 2 Feb 2026
Viewed by 148
Abstract
The detection of defects in Printed Circuit Boards (PCBs) is a critical yet challenging task in industrial quality control, characterized by the prevalence of small targets and complex backgrounds. While deep learning models like YOLOv5 have shown promise, they often lack the ability [...] Read more.
The detection of defects in Printed Circuit Boards (PCBs) is a critical yet challenging task in industrial quality control, characterized by the prevalence of small targets and complex backgrounds. While deep learning models like YOLOv5 have shown promise, they often lack the ability to quantify predictive uncertainty, leading to overconfident errors in challenging scenarios—a major source of false alarms and reduced reliability in automated manufacturing inspection lines. From a Bayesian perspective, this overconfidence signifies a failure in probabilistic calibration, which is crucial for trustworthy automated inspection. To address this, we propose MFE-YOLO, a Bayesian-enhanced detection framework built upon YOLOv5 that systematically integrates uncertainty-aware mechanisms to improve both accuracy and operational reliability in real-world settings. First, we construct a multi-background PCB defect dataset with diverse substrate colors and shapes, enhancing the model’s ability to generalize beyond the single-background bias of existing data. Second, we integrate the Convolutional Block Attention Module (CBAM), reinterpreted through a Bayesian lens as a feature-wise uncertainty weighting mechanism, to suppress background interference and amplify salient defect features. Third, we propose a novel FIoU loss function, redesigned within a probabilistic framework to improve bounding box regression accuracy and implicitly capture localization uncertainty, particularly for small defects. Extensive experiments demonstrate that MFE-YOLO achieves state-of-the-art performance, with mAP@0.5 and mAP@0.5:0.95 values of 93.9% and 59.6%, respectively, outperforming existing detectors, including YOLOv8 and EfficientDet. More importantly, the proposed framework yields better-calibrated confidence scores, significantly reducing false alarms and enabling more reliable human-in-the-loop verification. This work provides a deployable, uncertainty-aware solution for high-throughput PCB inspection, advancing toward trustworthy and efficient quality control in modern manufacturing environments. Full article
(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)
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44 pages, 29351 KB  
Article
Bayesian-Inspired Dynamic-Lag Causal Graphs and Role-Aware Transformers for Landslide Displacement Forecasting
by Fan Zhang, Yuanfa Ji, Xiaoming Liu, Siyuan Liu, Zhang Lu, Xiyan Sun, Shuai Ren and Xizi Jia
Entropy 2026, 28(1), 7; https://doi.org/10.3390/e28010007 - 20 Dec 2025
Viewed by 387
Abstract
Increasingly frequent intense rainfall is increasing landslide occurrence and risk. In southern China in particular, steep slopes and thin residual soils produce frequent landslide events with pronounced spatial heterogeneity. Therefore, displacement prediction methods that function across sites and deformation regimes in similar settings [...] Read more.
Increasingly frequent intense rainfall is increasing landslide occurrence and risk. In southern China in particular, steep slopes and thin residual soils produce frequent landslide events with pronounced spatial heterogeneity. Therefore, displacement prediction methods that function across sites and deformation regimes in similar settings are essential for early warning. Most existing approaches adopt a multistage pipeline that decomposes, predicts, and recombines, often leading to complex architectures with weak cross-domain transfer and limited adaptability. To address these limitations, we present CRAFormer, a causal role-aware Transformer guided by a dynamic-lag Bayesian network-style causal graph learned from historical observations. In our system, the discovered directed acyclic graph (DAG) partitions drivers into five causal roles and induces role-specific, non-anticipative masks for lightweight branch encoders, while a context-aware Top-2 gate sparsely fuses the branch outputs, yielding sample-wise attributions. To safely exploit exogenous rainfall forecasts, next-day rainfall is entered exclusively through an ICS tail with a leakage-free block mask, a non-negative readout, and a rainfall monotonicity regularizer. In this study, we curate two long-term GNSS datasets from Guangxi (LaMenTun and BaYiTun) that capture slow creep and step-like motions during extreme rainfall. Under identical inputs and a unified protocol, CRAFormer reduces the MAE and RMSE by 59–79% across stations relative to the strongest baseline, and it lowers magnitude errors near turning points and step events, demonstrating robust performance for two contrasting landslides within a shared regional setting. Ablations confirm the contributions of the DBN-style causal masks, the leakage-free ICS tail, and the monotonicity prior. These results highlight a practical path from causal discovery to forecast-compatible neural predictors for rainfall-induced landslides. Full article
(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)
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31 pages, 4844 KB  
Article
GAME-YOLO: Global Attention and Multi-Scale Enhancement for Low-Visibility UAV Detection with Sub-Pixel Localization
by Ruohai Di, Hao Fan, Yuanzheng Ma, Jinqiang Wang and Ruoyu Qian
Entropy 2025, 27(12), 1263; https://doi.org/10.3390/e27121263 - 18 Dec 2025
Viewed by 554
Abstract
Detecting low-altitude, slow-speed, small (LSS) UAVs is especially challenging in low-visibility scenes (low light, haze, motion blur), where inherent uncertainties in sensor data and object appearance dominate. We propose GAME-YOLO, a novel detector that integrates a Bayesian-inspired probabilistic reasoning framework with Global Attention [...] Read more.
Detecting low-altitude, slow-speed, small (LSS) UAVs is especially challenging in low-visibility scenes (low light, haze, motion blur), where inherent uncertainties in sensor data and object appearance dominate. We propose GAME-YOLO, a novel detector that integrates a Bayesian-inspired probabilistic reasoning framework with Global Attention and Multi-Scale Enhancement to improve small-object perception and sub-pixel-level localization. Built on YOLOv11, our framework comprises: (i) a visibility restoration front-end that probabilistically infers and enhances latent image clarity; (ii) a global-attention-augmented backbone that performs context-aware feature selection; (iii) an adaptive multi-scale fusion neck that dynamically weights feature contributions; (iv) a sub-pixel-aware small-object detection head (SOH) that leverages high-resolution feature grids to model sub-pixel offsets; and (v) a novel Shape-Aware IoU loss combined with focal loss. Extensive experiments on the LSS2025-DET dataset demonstrate that GAME-YOLO achieves state-of-the-art performance, with an AP@50 of 52.0% and AP@[0.50:0.95] of 32.0%, significantly outperforming strong baselines such as LEAF-YOLO (48.3% AP@50) and YOLOv11 (36.2% AP@50). The model maintains high efficiency, operating at 48 FPS with only 7.6 M parameters and 19.6 GFLOPs. Ablation studies confirm the complementary gains from our probabilistic design choices, including a +10.5 pp improvement in AP@50 over the baseline. Cross-dataset evaluation on VisDrone-DET2021 further validates its generalization capability, achieving 39.2% AP@50. These results indicate that GAME-YOLO offers a practical and reliable solution for vision-based UAV surveillance by effectively marrying the efficiency of deterministic detectors with the robustness principles of Bayesian inference. Full article
(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)
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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 608
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 1077
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 1177
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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