{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T05:26:11Z","timestamp":1764739571770,"version":"3.46.0"},"reference-count":50,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T00:00:00Z","timestamp":1764288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China General Project","award":["62376089"],"award-info":[{"award-number":["62376089"]}]},{"name":"Yellow Crane Talents Program funding and Research Funding from Hubei University of Technology","award":["HBUT 4301\/00550"],"award-info":[{"award-number":["HBUT 4301\/00550"]}]},{"name":"Key Research and Development Program of Hubei Province","award":["2023BEB024"],"award-info":[{"award-number":["2023BEB024"]}]},{"name":"Start-up Research Funding from Hubei University of Technology","award":["HBUT 4301\/00801"],"award-info":[{"award-number":["HBUT 4301\/00801"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Tree-Augmented Naive Bayes (TAN) is an interpretable graphical structure model. However, its structure learning for continuous attributes depends on the class-conditional mutual information, which is sensitive to one-dimensional or two-dimensional density estimation. Accurate estimation is challenging under complex distributions such as multi-peak, long-tailed and heteroscedastic cases. To address this issue, we propose a structure learning method for TAN based on Fast Generative Bootstrap Maximum Likelihood Estimation (TAN-FGBMLE). FGBMLE consists of two stages of work. In the first stage, resampling weights and random noise are input into a network generator to rapidly produce candidate parameters, efficiently covering the latent density space without repeated independent optimization. In the second stage, optimal mixture weights are estimated by maximum likelihood estimation, assigning appropriate contributions to each candidate component. This design enables fast and accurate complex density estimation for both single and joint attributes, providing reliable computation of class-conditional mutual information. The TAN structure is then constructed using Prim\u2019s maximum spanning tree algorithm. Experiments show that our estimation method attains higher fitting accuracy and lower runtime compared with traditional nonparametric estimators. By using open-source datasets, the TAN-FGBMLE achieves superior accuracy and recall compared to classic methods, demonstrating good robustness and interpretability. On publicly available real air quality data, it has a high classification result and produces graph structures that more accurately capture dependencies among continuous attributes.<\/jats:p>","DOI":"10.3390\/e27121216","type":"journal-article","created":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T09:34:37Z","timestamp":1764581677000},"page":"1216","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["TAN-FGBMLE: Tree-Augmented Naive Bayes Structure Learning Based on Fast Generative Bootstrap Maximum Likelihood Estimation for Continuous-Variable Classification"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-6172-0486","authenticated-orcid":false,"given":"Chenghao","family":"Wei","sequence":"first","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"},{"name":"Hubei Provincial Key Laboratory of Green Intelligent Computing Power Network, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianyu","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"},{"name":"Hubei Provincial Key Laboratory of Green Intelligent Computing Power Network, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7342-0677","authenticated-orcid":false,"given":"Chen","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"},{"name":"Hubei Provincial Key Laboratory of Green Intelligent Computing Power Network, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pukai","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"},{"name":"Hubei Provincial Key Laboratory of Green Intelligent Computing Power Network, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiwei","family":"Ye","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"},{"name":"Hubei Provincial Key Laboratory of Green Intelligent Computing Power Network, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. 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