{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T07:07:36Z","timestamp":1774422456296,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,22]],"date-time":"2023-07-22T00:00:00Z","timestamp":1689984000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Faculty of Transportation Sciences, Czech Technical University in Prague\u2014Future Fund"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The article presents a novel algorithm for merging Bayesian networks generated by different methods, such as expert knowledge and data-driven approaches, while leveraging a symmetry-based approach. The algorithm combines the strengths of each input network to create a more comprehensive and accurate network. Evaluations on traffic accident data from Prague in the Czech Republic and accidents on railway crossings demonstrate superior predictive performance, as measured by prediction error metric. The algorithm identifies and incorporates symmetric nodes into the final network, ensuring consistent representations across different methods. The merged network, incorporating nodes selected from both the expert and algorithm networks, provides a more comprehensive and accurate representation of the relationships among variables in the dataset. Future research could focus on extending the algorithm to deal with cycles and improving the handling of conditional probability tables. Overall, the proposed algorithm demonstrates the effectiveness of combining different sources of knowledge in Bayesian network modeling.<\/jats:p>","DOI":"10.3390\/sym15071461","type":"journal-article","created":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:57:30Z","timestamp":1690160250000},"page":"1461","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Novel Algorithm for Merging Bayesian Networks"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7589-6206","authenticated-orcid":false,"given":"Miroslav","family":"Vani\u0161","sequence":"first","affiliation":[{"name":"Faculty of Transportation Sciences, Czech Technical University in Prague, 110 00 Prague, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0624-0430","authenticated-orcid":false,"given":"Zden\u011bk","family":"Lokaj","sequence":"additional","affiliation":[{"name":"Faculty of Transportation Sciences, Czech Technical University in Prague, 110 00 Prague, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0049-4381","authenticated-orcid":false,"given":"Martin","family":"\u0160rot\u00fd\u0159","sequence":"additional","affiliation":[{"name":"Faculty of Transportation Sciences, Czech Technical University in Prague, 110 00 Prague, Czech Republic"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,22]]},"reference":[{"key":"ref_1","unstructured":"Glymour, M., Pearl, J., and Jewell, N.P. (2016). Causal Inference in Statistics: A Primer, John Wiley & Sons."},{"key":"ref_2","unstructured":"Koller, D., and Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques, MIT Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s13748-019-00194-y","article-title":"A survey on Bayesian network structure learning from data","volume":"8","author":"Scanagatta","year":"2019","journal-title":"Prog. Artif. Intell."},{"key":"ref_4","unstructured":"Kj\u00e6rulff, U.B., and Madsen, A.L. (2005). Probabilistic Networks\u2014An Introduction to Bayesian Networks and Influence Diagrams, Aalborg University."},{"key":"ref_5","unstructured":"Wasserman, L. (2013). All of Statistics: A Concise Course in Statistical Inference, Springer."},{"key":"ref_6","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Darwiche, A. (2009). Modeling and Reasoning with Bayesian Networks, Cambridge University Press.","DOI":"10.1017\/CBO9780511811357"},{"key":"ref_8","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":"ref_9","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Spirtes, P., Glymour, C., and Scheines, R. (2000). Causation, Prediction, and Search, MIT Press.","DOI":"10.7551\/mitpress\/1754.001.0001"},{"key":"ref_11","first-page":"507","article-title":"Optimal structure identification with greedy search","volume":"3","author":"Chickering","year":"2003","journal-title":"J. Mach. Learn. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1346","DOI":"10.1111\/risa.13798","article-title":"Bayesian network applications for sustainable holistic water resources management: Modeling opportunities for South Africa","volume":"42","author":"Govender","year":"2022","journal-title":"Risk Anal."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Li, H., Yazdi, M., Huang, H.Z., Huang, C.G., Peng, W., Nedjati, A., and Adesina, K.A. (2023). A fuzzy rough copula Bayesian network model for solving complex hospital service quality assessment. Complex Intell. Syst.","DOI":"10.1007\/s40747-023-01002-w"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1002\/int.10086","article-title":"Qualitative combination of Bayesian networks","volume":"18","author":"Moral","year":"2003","journal-title":"Int. J. Intell. Syst."},{"key":"ref_15","unstructured":"Jiang, C.a., Leong, T.Y., and Kim-Leng, P. (2005, January 22\u201326). PGMC: A framework for probabilistic graphical model combination. Proceedings of the American Medical Informatics Association Annual Symposium, Washington, DC, USA."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1016\/j.patcog.2013.12.005","article-title":"A novel method for combining Bayesian networks, theoretical analysis, and its applications","volume":"47","author":"Feng","year":"2014","journal-title":"Pattern Recognit."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.knosys.2019.03.014","article-title":"An analytical threshold for combining bayesian networks","volume":"175","author":"Gross","year":"2019","journal-title":"Knowl. Based Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1109\/34.58871","article-title":"Neural network ensembles","volume":"12","author":"Hansen","year":"1990","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1613\/jair.614","article-title":"Popular ensemble methods: An empirical study","volume":"11","author":"Opitz","year":"1999","journal-title":"J. Artif. Intell. Res."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Polikar, R. (2012). Ensemble Machine Learning: Methods and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-9326-7_1"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1007\/s11222-019-09857-1","article-title":"Learning Bayesian networks from big data with greedy search: Computational complexity and efficient implementation","volume":"29","author":"Scutari","year":"2019","journal-title":"Stat. Comput."},{"key":"ref_22","unstructured":"Kareem, S., and Okur, M.C. (2018). Bayesian Network Structure Learning Using Hybrid Bee Optimization and Greedy Search, \u00c7ukurova University."},{"key":"ref_23","unstructured":"Vani\u0161, M. (2021). Optimization of Bayesian Networks and Their Prediction Properties. [Ph.D. Thesis, Czech Technical University in Prague]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1137\/0201010","article-title":"Depth-first search and linear graph algorithms","volume":"1","author":"Tarjan","year":"1972","journal-title":"SIAM J. Comput."},{"key":"ref_25","unstructured":"Meek, C. (1995, January 18\u201320). Causal inference and causal explanation with background knowledge. Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QC, Canada."},{"key":"ref_26","unstructured":"Wiecek, W., Bois, F.Y., and Gayraud, G. (2019). Structure learning of Bayesian networks involving cyclic structures. arXiv."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Korb, K.B., and Nicholson, A.E. (2010). Bayesian Artificial Intelligence, CRC Press.","DOI":"10.1201\/b10391"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc. Ser. (Methodol.)"},{"key":"ref_29","unstructured":"Casella, G., and Berger, R.L. (2021). Statistical Inference, Cengage Learning."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Vani\u0161, M., and Urbaniec, K. (2016, January 26\u201327). Employing Bayesian Networks and conditional probability functions for determining dependences in road traffic accidents data. Proceedings of the 2017 Smart City Symposium Prague (SCSP), Prague, Czech Republic.","DOI":"10.1109\/SCSP.2017.7973842"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1461\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:17:10Z","timestamp":1760127430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1461"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,22]]},"references-count":30,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071461"],"URL":"https:\/\/doi.org\/10.3390\/sym15071461","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,22]]}}}