{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T21:01:52Z","timestamp":1784667712472,"version":"3.55.0"},"reference-count":50,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T00:00:00Z","timestamp":1784160000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Approximate Reasoning"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.ijar.2026.109771","type":"journal-article","created":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T15:05:10Z","timestamp":1784214310000},"page":"109771","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["A BGe score for tied-covariance mixture Gaussian Bayesian networks"],"prefix":"10.1016","volume":"198","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2604-9270","authenticated-orcid":false,"given":"Marco","family":"Grzegorczyk","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.ijar.2026.109771_bib0001","series-title":"Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference","author":"Pearl","year":"1988"},{"key":"10.1016\/j.ijar.2026.109771_bib0002","series-title":"Probabilistic Reasoning in Expert Systems: Theory and Algorithms","author":"Neapolitan","year":"1989"},{"key":"10.1016\/j.ijar.2026.109771_bib0003","article-title":"Probabilistic Graphical Models: Principles and Techniques","author":"Koller","year":"2009"},{"key":"10.1016\/j.ijar.2026.109771_bib0004","series-title":"Bayesian Networks: With Examples in R","author":"Scutari","year":"2014"},{"key":"10.1016\/j.ijar.2026.109771_bib0005","series-title":"Learning from Data: Artificial Intelligence and Statistics","first-page":"121","article-title":"Learning Bayesian networks is NP-complete","volume":"5","author":"Chickering","year":"1996"},{"key":"10.1016\/j.ijar.2026.109771_bib0006","series-title":"International Conference on Probabilistic Graphical Models","first-page":"416","article-title":"Who learns better Bayesian network structures: constraint-based, score-based or hybrid algorithms?","author":"Scutari","year":"2018"},{"key":"10.1016\/j.ijar.2026.109771_bib0007","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.ijar.2019.10.003","article-title":"Who learns better Bayesian network structures: accuracy and speed of structure learning algorithms","volume":"115","author":"Scutari","year":"2019","journal-title":"Int. J. Approx. Reason."},{"key":"10.1016\/j.ijar.2026.109771_bib0008","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1111\/stan.12197","article-title":"Bayesian network models for incomplete and dynamic data","volume":"74","author":"Scutari","year":"2020","journal-title":"Stat. Neerl."},{"key":"10.1016\/j.ijar.2026.109771_bib0009","doi-asserted-by":"crossref","first-page":"8721","DOI":"10.1007\/s10462-022-10351-w","article-title":"A survey of Bayesian network structure learning","volume":"56","author":"Kitson","year":"2023","journal-title":"Artif. Intell. Rev."},{"key":"10.1016\/j.ijar.2026.109771_bib0010","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.ijar.2022.09.016","article-title":"Effective and efficient structure learning with pruning and model averaging strategies","volume":"151","author":"Constantinou","year":"2022","journal-title":"Int. J. Approx. Reason."},{"key":"10.1016\/j.ijar.2026.109771_bib0011","doi-asserted-by":"crossref","first-page":"215","DOI":"10.2307\/1403615","article-title":"Bayesian graphical models for discrete data","volume":"63","author":"Madigan","year":"1995","journal-title":"Int. Stat. Rev."},{"key":"10.1016\/j.ijar.2026.109771_bib0012","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1023\/A:1020202028934","article-title":"Improving Markov chain Monte Carlo model search for data mining","volume":"50","author":"Giudici","year":"2003","journal-title":"Mach. Learn."},{"key":"10.1016\/j.ijar.2026.109771_bib0013","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1023\/A:1020249912095","article-title":"Being Bayesian about network structure: a Bayesian approach to structure discovery in Bayesian networks","volume":"50","author":"Friedman","year":"2003","journal-title":"Mach. Learn."},{"issue":"2-3","key":"10.1016\/j.ijar.2026.109771_bib0014","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s10994-008-5057-7","article-title":"Improving the structure MCMC sampler for Bayesian networks by introducing a new edge reversal move","volume":"71","author":"Grzegorczyk","year":"2008","journal-title":"Mach. Learn."},{"key":"10.1016\/j.ijar.2026.109771_bib0015","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1080\/01621459.2015.1133426","article-title":"Partition MCMC for inference on acyclic digraphs","volume":"112","author":"Kuipers","year":"2017","journal-title":"J. Am. Stat. Assoc."},{"key":"10.1016\/j.ijar.2026.109771_bib0016","unstructured":"D. Nikzad, A. Zhilkin, J. Harviainen, J. Kuipers, G. Moffa, M. Koivisto, Scaling Up Bayesian DAG Sampling, 2025, arXiv: 2510.25254[cs.LG]."},{"key":"10.1016\/j.ijar.2026.109771_bib0017","series-title":"Proceedings of the Fourteenth International Conference on Machine Learning (ICML)","first-page":"125","article-title":"Learning belief networks in the presence of missing values and hidden variables","author":"Friedman","year":"1997"},{"issue":"12","key":"10.1016\/j.ijar.2026.109771_bib0018","doi-asserted-by":"crossref","DOI":"10.3390\/a13120329","article-title":"Hard and soft EM in Bayesian network learning from incomplete data","volume":"13","author":"Ruggieri","year":"2020","journal-title":"Algorithms"},{"key":"10.1016\/j.ijar.2026.109771_bib0019","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.ijar.2021.07.015","article-title":"Learning Bayesian networks from incomplete data with the node-average likelihood","volume":"138","author":"Bodewes","year":"2021","journal-title":"Int. J. Approx. Reason."},{"key":"10.1016\/j.ijar.2026.109771_bib0020","first-page":"1","article-title":"Learning Bayesian networks from ordinal data","volume":"22","author":"Luo","year":"2021","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.ijar.2026.109771_bib0021","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1007\/s11336-024-09969-2","article-title":"Learning Bayesian networks: a copula approach for mixed-type data","volume":"89","author":"Castelletti","year":"2024","journal-title":"Psychometrika"},{"key":"10.1016\/j.ijar.2026.109771_bib0022","unstructured":"M. Scauda, J. Kuipers, G. Moffa, A Latent Causal Inference Framework for Ordinal Variables, 2025, arXiv: 2502.10276[stat.ME]."},{"key":"10.1016\/j.ijar.2026.109771_bib0023","first-page":"112","article-title":"Learning Bayesian networks: search methods and experimental results","volume":"20","author":"Chickering","year":"1995","journal-title":"Mach. Learn."},{"key":"10.1016\/j.ijar.2026.109771_bib0024","doi-asserted-by":"crossref","DOI":"10.1016\/j.ijar.2023.108954","article-title":"Being Bayesian about learning Gaussian Bayesian networks from incomplete data","volume":"160","author":"Grzegorczyk","year":"2023","journal-title":"Int. J. Approx. Reason."},{"key":"10.1016\/j.ijar.2026.109771_bib0025","doi-asserted-by":"crossref","DOI":"10.1016\/j.ijar.2024.109205","article-title":"Being Bayesian about learning Bayesian networks from ordinal data","volume":"170","author":"Grzegorczyk","year":"2024","journal-title":"Int. J. Approx. Reason."},{"key":"10.1016\/j.ijar.2026.109771_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.ijar.2025.109549","article-title":"Being Bayesian about learning Bayesian networks from hybrid data","volume":"187","author":"Grzegorczyk","year":"2025","journal-title":"Int. J. Approx. Reason."},{"key":"10.1016\/j.ijar.2026.109771_bib0027","series-title":"BIBM International Conference on Bioinformatics and Biomedicine","first-page":"362","article-title":"Inference of gene pathways using gaussian mixture models","author":"Ko","year":"2007"},{"issue":"18","key":"10.1016\/j.ijar.2026.109771_bib0028","doi-asserted-by":"crossref","first-page":"2071","DOI":"10.1093\/bioinformatics\/btn367","article-title":"Modelling non-stationary gene regulatory processes with a non-homogeneous Bayesian network and the allocation sampler","volume":"24","author":"Grzegorczyk","year":"2008","journal-title":"Bioinformatics"},{"issue":"2","key":"10.1016\/j.ijar.2026.109771_bib0029","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s11222-006-9014-7","article-title":"Bayesian finite mixtures with an unknown number of components: the allocation sampler","volume":"17","author":"Nobile","year":"2007","journal-title":"Stat. Comput."},{"key":"10.1016\/j.ijar.2026.109771_bib0030","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"10.1016\/j.ijar.2026.109771_bib0031","doi-asserted-by":"crossref","unstructured":"M.E.J. Newman, Fast sampling and model selection for Bayesian mixture models, 2025, arXiv: 2501.07668[stat.ML].","DOI":"10.1007\/s11222-025-10753-0"},{"issue":"5","key":"10.1016\/j.ijar.2026.109771_bib0032","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1016\/0031-3203(94)00125-6","article-title":"Gaussian parsimonious clustering models","volume":"28","author":"Celeux","year":"1995","journal-title":"Pattern Recognit."},{"issue":"3","key":"10.1016\/j.ijar.2026.109771_bib0033","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1109\/34.990138","article-title":"Unsupervised learning of finite mixture models","volume":"24","author":"Figueiredo","year":"2002","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"2","key":"10.1016\/j.ijar.2026.109771_bib0034","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s13042-023-01910-w","article-title":"Gaussian mixture model with local consistency: a hierarchical minimum message length-based approach","volume":"15","author":"Li","year":"2024","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"10.1016\/j.ijar.2026.109771_bib0035","first-page":"2399","article-title":"Manifold regularization: a geometric framework for learning from labeled and unlabeled examples","volume":"7","author":"Belkin","year":"2006","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.ijar.2026.109771_bib0036","series-title":"Proceedings of the 18th International Conference on Intelligent Systems and Knowledge Engineering (ISKE)","first-page":"47","article-title":"Gaussian function representation of 2nd-order normal cloud model","author":"Li","year":"2023"},{"key":"10.1016\/j.ijar.2026.109771_bib0037","first-page":"445","article-title":"Learning equivalence classes of Bayesian-network structures","volume":"2","author":"Chickering","year":"2002","journal-title":"J. Mach. Learn. Res."},{"key":"10.1016\/j.ijar.2026.109771_bib0038","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1023\/A:1022623210503","article-title":"Learning Bayesian networks: the combination of knowledge and statistical data","volume":"20","author":"Heckerman","year":"1995","journal-title":"Mach. Learn."},{"key":"10.1016\/j.ijar.2026.109771_bib0039","series-title":"Proceedings of the 11th Annual Conference on Uncertainty in Artificial Intelligence (UAI-95)","first-page":"274","article-title":"Learning Bayesian networks: a unification for discrete and Gaussian domains","author":"Heckerman","year":"1995"},{"key":"10.1016\/j.ijar.2026.109771_bib0040","series-title":"Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence","first-page":"235","article-title":"Learning Gaussian networks","author":"Geiger","year":"1994"},{"issue":"5","key":"10.1016\/j.ijar.2026.109771_bib0041","doi-asserted-by":"crossref","first-page":"1412","DOI":"10.1214\/aos\/1035844981","article-title":"Parameter priors for directed acyclic graphical models and the characterization of several probability distributions","volume":"30","author":"Geiger","year":"2002","journal-title":"Ann. Stat."},{"issue":"4","key":"10.1016\/j.ijar.2026.109771_bib0042","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1214\/14-AOS1217","article-title":"Addendum on the scoring of Gaussian directed acyclic graphical models","volume":"42","author":"Kuipers","year":"2014","journal-title":"Ann. Stat."},{"key":"10.1016\/j.ijar.2026.109771_bib0043","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1287\/mnsc.35.5.527","article-title":"Gaussian influence diagrams","volume":"35","author":"Shachter","year":"1989","journal-title":"Manag. Sci."},{"key":"10.1016\/j.ijar.2026.109771_bib0044","series-title":"Technical Report","article-title":"Fish Market Dataset","author":"Aung Pyae","year":"2019"},{"issue":"2","key":"10.1016\/j.ijar.2026.109771_bib0045","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1111\/j.1469-1809.1936.tb02137.x","article-title":"The use of multiple measurements in taxonomic problems","volume":"7","author":"Fisher","year":"1936","journal-title":"Ann. Eugen."},{"key":"10.1016\/j.ijar.2026.109771_bib0046","first-page":"189","article-title":"Multivariate data analysis as a discriminating method of the origin of wines","volume":"25","author":"Forina","year":"1986","journal-title":"Vitis"},{"key":"10.1016\/j.ijar.2026.109771_bib0047","series-title":"Information Technologies in Biomedicine","first-page":"15","article-title":"Complete gradient clustering algorithm for features analysis of X-ray images","volume":"69","author":"Charytanowicz","year":"2010"},{"key":"10.1016\/j.ijar.2026.109771_bib0048","series-title":"Proceedings of the 11th International Conference on Probabilistic Graphical Models","first-page":"73","article-title":"Using mixed-effects models to learn Bayesian networks from related data sets","volume":"186","author":"Scutari","year":"2022"},{"key":"10.1016\/j.ijar.2026.109771_bib0049","series-title":"Optimal Statistical Decisions","author":"DeGroot","year":"1970"},{"key":"10.1016\/j.ijar.2026.109771_bib0050","series-title":"Pattern Recognition and Machine Learning","author":"Bishop","year":"2006"}],"container-title":["International Journal of Approximate Reasoning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0888613X26001465?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0888613X26001465?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T20:09:22Z","timestamp":1784664562000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0888613X26001465"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":50,"alternative-id":["S0888613X26001465"],"URL":"https:\/\/doi.org\/10.1016\/j.ijar.2026.109771","relation":{},"ISSN":["0888-613X"],"issn-type":[{"value":"0888-613X","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"A BGe score for tied-covariance mixture Gaussian Bayesian networks","name":"articletitle","label":"Article Title"},{"value":"International Journal of Approximate Reasoning","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.ijar.2026.109771","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Author(s). Published by Elsevier Inc.","name":"copyright","label":"Copyright"}],"article-number":"109771"}}