{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T15:45:34Z","timestamp":1783611934037,"version":"3.55.0"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T00:00:00Z","timestamp":1754092800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T00:00:00Z","timestamp":1754092800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Northeastern University USA"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Intell Manuf"],"published-print":{"date-parts":[[2026,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Causal inference has recently gained notable attention across various fields like biology, healthcare, and environmental science, especially within explainable artificial intelligence (xAI) systems, for uncovering the causal relationships among multiple variables and outcomes. Yet, it has not been fully recognized and deployed in the manufacturing systems. In this paper, we introduce a scalable, and flexible Bayesian federated learning framework, xFBCI, designed to explore causality through treatment effect estimation in distributed manufacturing systems. By leveraging federated Bayesian learning, we efficiently estimate posterior of local parameters to derive the propensity score for each client without accessing local private data. These scores are then used to estimate the treatment effect using propensity score matching (PSM). Through simulations on various datasets and real-world Electrohydrodynamic (EHD) printing data and smart manufacturing maintenance data, we demonstrate that our approach outperforms standard Bayesian causal inference methods and several state-of-the-art federated learning benchmarks.<\/jats:p>","DOI":"10.1007\/s10845-025-02665-7","type":"journal-article","created":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T15:00:32Z","timestamp":1754146832000},"page":"2807-2831","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bayesian federated causal inference and its application in manufacturing"],"prefix":"10.1007","volume":"37","author":[{"given":"Xiaofeng","family":"Xiao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Khawlah","family":"Alharbi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pengyu","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hantang","family":"Qin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9929-8895","authenticated-orcid":false,"given":"Xubo","family":"Yue","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,8,2]]},"reference":[{"key":"2665_CR1","unstructured":"McMahan, B., Moore, E., Ramage, D., Hampson, S., Arcas, B.A. ( 2017). Communication-efficient learning of deep networks from decentralized data. In: Artificial Intelligence and Statistics, pp. 1273\u2013 1282. PMLR"},{"key":"2665_CR2","unstructured":"Jordan, M.I., Lee, J.D., Yang, Y. (2019) Communication-efficient distributed statistical inference. Journal of the American Statistical Association"},{"key":"2665_CR3","doi-asserted-by":"publisher","first-page":"156071","DOI":"10.1109\/ACCESS.2021.3127448","volume":"9","author":"R Kontar","year":"2021","unstructured":"Kontar, R., Shi, N., Yue, X., Chung, S., Byon, E., Chowdhury, M., Jin, J., Kontar, W., Masoud, N., Nouiehed, M., et al. (2021). The internet of federated things (IOFT). IEEE Access, 9, 156071\u2013156113.","journal-title":"IEEE Access"},{"key":"2665_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s41666-020-00082-4","volume":"5","author":"J Xu","year":"2021","unstructured":"Xu, J., Glicksberg, B. S., Su, C., Walker, P., Bian, J., & Wang, F. (2021). Federated learning for healthcare informatics. Journal of healthcare informatics research, 5, 1\u201319.","journal-title":"Journal of healthcare informatics research"},{"issue":"11","key":"2665_CR5","doi-asserted-by":"publisher","first-page":"8229","DOI":"10.1109\/JIOT.2022.3150363","volume":"9","author":"B Ghimire","year":"2022","unstructured":"Ghimire, B., & Rawat, D. B. (2022). Recent advances on federated learning for cybersecurity and cybersecurity for federated learning for internet of things. IEEE Internet Things Journal, 9(11), 8229\u20138249.","journal-title":"IEEE Internet Things J"},{"key":"2665_CR6","doi-asserted-by":"crossref","unstructured":"Pearl, J. (2010). Causal inference. Causality: objectives and assessment, 39\u201358","DOI":"10.1017\/CBO9780511803161"},{"key":"2665_CR7","doi-asserted-by":"crossref","unstructured":"Rubin, D.B.( 1997). Estimating causal effects from large data sets using propensity scores. Annals of internal medicine 127( 8_Part_2), 757\u2013 763","DOI":"10.7326\/0003-4819-127-8_Part_2-199710151-00064"},{"issue":"7","key":"2665_CR8","doi-asserted-by":"publisher","first-page":"578","DOI":"10.1136\/jech.2004.029496","volume":"60","author":"MA Hern\u00e1n","year":"2006","unstructured":"Hern\u00e1n, M. A., & Robins, J. M. (2006). Estimating causal effects from epidemiological data. Journal of Epidemiology & Community Health, 60(7), 578\u2013586.","journal-title":"Journal of Epidemiology & Community Health"},{"key":"2665_CR9","doi-asserted-by":"publisher","first-page":"16261","DOI":"10.52202\/068431-1183","volume":"35","author":"C Toth","year":"2022","unstructured":"Toth, C., Lorch, L., Knoll, C., Krause, A., Pernkopf, F., Peharz, R., & Von K\u00fcgelgen, J. (2022). Active bayesian causal inference. Advances in Neural Information Processing Systems, 35, 16261\u201316275.","journal-title":"Adv. Neural. Inf. Process. Syst."},{"issue":"2247","key":"2665_CR10","doi-asserted-by":"publisher","first-page":"20220153","DOI":"10.1098\/rsta.2022.0153","volume":"381","author":"F Li","year":"2023","unstructured":"Li, F., Ding, P., & Mealli, F. (2023). Bayesian causal inference: a critical review. Philosophical Transactions of the Royal Society A, 381(2247), 20220153.","journal-title":"Phil. Trans. R. Soc. A"},{"key":"2665_CR11","unstructured":"Vo, T.V., Leong, T.-Y., et al. (2023). Federated causal inference from observational data. arXiv preprint arXiv:2308.13047"},{"issue":"24","key":"2665_CR12","doi-asserted-by":"publisher","first-page":"4418","DOI":"10.1002\/sim.9868","volume":"42","author":"R Xiong","year":"2023","unstructured":"Xiong, R., Koenecke, A., Powell, M., Shen, Z., Vogelstein, J. T., & Athey, S. (2023). Federated causal inference in heterogeneous observational data. Statistics in Medicine, 42(24), 4418\u20134439.","journal-title":"Stat. Med."},{"key":"2665_CR13","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.inffus.2021.11.003","volume":"81","author":"Y-L Chou","year":"2022","unstructured":"Chou, Y.-L., Moreira, C., Bruza, P., Ouyang, C., & Jorge, J. (2022). Counterfactuals and causability in explainable artificial intelligence: Theory, algorithms, and applications. Information Fusion, 81, 59\u201383.","journal-title":"Information Fusion"},{"issue":"8","key":"2665_CR14","doi-asserted-by":"publisher","first-page":"5031","DOI":"10.1109\/TII.2022.3146552","volume":"18","author":"I Ahmed","year":"2022","unstructured":"Ahmed, I., Jeon, G., & Piccialli, F. (2022). From artificial intelligence to explainable artificial intelligence in industry 4.0: a survey on what, how, and where. EEE transactions on industrial informatics, 18(8), 5031\u20135042.","journal-title":"IEEE Trans. Industr. Inf."},{"key":"2665_CR15","doi-asserted-by":"crossref","unstructured":"Wolf, R., Jiang, L., Alharbi, K., Zhang, P., Wang, C., Qin, H.(2024). Heterogeneous transfer learning of electrohydrodynamic printing under zero-gravity toward in-space manufacturing. Journal of Manufacturing Science and Engineering 146(12)","DOI":"10.1115\/1.4066097"},{"key":"2665_CR16","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1016\/j.mfglet.2024.09.029","volume":"41","author":"X Jiang","year":"2024","unstructured":"Jiang, X., Zhang, P., & Qin, H. (2024). A qualitative validation of an in-situ monitoring system for EHD inkjet printing via laser diffraction. Manufacturing Letters, 41, 248\u2013252.","journal-title":"Manufacturing Letters"},{"key":"2665_CR17","doi-asserted-by":"crossref","unstructured":"Jiang, L., Wolf, R., Alharbi, K., Qin, H. (2024). In situ monitoring and recognition of printing quality in electrohydrodynamic inkjet printing via machine learning. Journal of Manufacturing Science and Engineering 146(11)","DOI":"10.1115\/1.4066124"},{"key":"2665_CR18","unstructured":"Almod\u00f3var, A., Parras, J., Zazo, S.(2023). Federated learning for causal inference using deep generative disentangled models. In: Deep Generative Models for Health Workshop NeurIPS 2023"},{"key":"2665_CR19","unstructured":"Khellaf, R., Bellet, A., Josse, J.(2024). Federated causal inference: Multi-centric ate estimation beyond meta-analysis. arXiv preprint arXiv:2410.16870"},{"key":"2665_CR20","doi-asserted-by":"publisher","first-page":"106854","DOI":"10.1016\/j.cie.2020.106854","volume":"149","author":"L Li","year":"2020","unstructured":"Li, L., Fan, Y., Tse, M., & Lin, K.-Y. (2020). A review of applications in federated learning. Computers & Industrial Engineering, 149, 106854.","journal-title":"Computers & Industrial Engineering"},{"key":"2665_CR21","unstructured":"Cao, L., Chen, H., Fan, X., Gama, J., Ong, Y.-S., Kumar, V.(2023). Bayesian federated learning: A survey. arXiv preprint arXiv:2304.13267"},{"key":"2665_CR22","unstructured":"Al-Shedivat, M., Gillenwater, J., Xing, E., Rostamizadeh, A.(2020). Federated learning via posterior averaging: A new perspective and practical algorithms. arXiv preprint arXiv:2010.05273"},{"key":"2665_CR23","unstructured":"Chen, H.-Y., Chao, W.-L.(2020). Fedbe: Making bayesian model ensemble applicable to federated learning. arXiv preprint arXiv:2009.01974"},{"key":"2665_CR24","doi-asserted-by":"publisher","first-page":"8687","DOI":"10.52202\/068431-0632","volume":"35","author":"N Kotelevskii","year":"2022","unstructured":"Kotelevskii, N., Vono, M., Durmus, A., & Moulines, E. (2022). Fedpop: A bayesian approach for personalised federated learning. Advances in Neural Information Processing Systems, 35, 8687\u20138701.","journal-title":"Adv. Neural. Inf. Process. Syst."},{"issue":"7","key":"2665_CR25","doi-asserted-by":"publisher","first-page":"3025","DOI":"10.1007\/s10845-022-01968-3","volume":"34","author":"T Deng","year":"2023","unstructured":"Deng, T., Li, Y., Liu, X., & Wang, L. (2023). Federated learning-based collaborative manufacturing for complex parts. Journal of Intelligent Manufacturing, 34(7), 3025\u20133038.","journal-title":"J. Intell. Manuf."},{"key":"2665_CR26","doi-asserted-by":"publisher","first-page":"57585","DOI":"10.1109\/ACCESS.2023.3282316","volume":"11","author":"I Kavasidis","year":"2023","unstructured":"Kavasidis, I., Lallas, E., Mountzouris, G., Gerogiannis, V. C., & Karageorgos, A. (2023). A federated learning framework for enforcing traceability in manufacturing processes. IEEE Access, 11, 57585\u201357597.","journal-title":"IEEE Access"},{"key":"2665_CR27","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1016\/j.future.2024.03.043","volume":"157","author":"H Liu","year":"2024","unstructured":"Liu, H., Li, S., Li, W., & Sun, W. (2024). Efficient decentralized optimization for edge-enabled smart manufacturing: A federated learning-based framework. Future Generation Computer Systems, 157, 422\u2013435.","journal-title":"Futur. Gener. Comput. Syst."},{"key":"2665_CR28","doi-asserted-by":"crossref","unstructured":"Thakur, A., Sindhwani, N.(2025). Intelligent load migration using federated learning in intelligent manufacturing. In: Intelligent Manufacturing and Industry 4.0, pp. 81\u2013 124. CRC Press.","DOI":"10.1201\/9781032630748-5"},{"issue":"469","key":"2665_CR29","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1198\/016214504000001880","volume":"100","author":"DB Rubin","year":"2005","unstructured":"Rubin, D. B. (2005). Causal inference using potential outcomes: Design, modeling, decisions. Journal of the American statistical Association, 100(469), 322\u2013331.","journal-title":"J. Am. Stat. Assoc."},{"issue":"1","key":"2665_CR30","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1198\/jcgs.2010.08162","volume":"20","author":"JL Hill","year":"2011","unstructured":"Hill, J. L. (2011). Bayesian nonparametric modeling for causal inference. Journal of Computational and Graphical Statistics, 20(1), 217\u2013240.","journal-title":"J. Comput. Graph. Stat."},{"key":"2665_CR31","doi-asserted-by":"publisher","first-page":"24459","DOI":"10.52202\/068431-1776","volume":"35","author":"TV Vo","year":"2022","unstructured":"Vo, T. V., Bhattacharyya, A., Lee, Y., & Leong, T.-Y. (2022). An adaptive kernel approach to federated learning of heterogeneous causal effects. Advances in Neural Information Processing Systems, 35, 24459\u201324473.","journal-title":"Adv. Neural. Inf. Process. Syst."},{"key":"2665_CR32","doi-asserted-by":"crossref","unstructured":"Kayaalp, M., Inan, Y., Koivunen, V., Sayed, A.H.(2024). Causal influence in federated edge inference. IEEE Transactions on Signal Processing","DOI":"10.1109\/TSP.2024.3507715"},{"key":"2665_CR33","doi-asserted-by":"crossref","unstructured":"Mu, J., Kadoch, M., Yuan, T., Lv, W., Liu, Q., Li, B.(2024). Explainable federated medical image analysis through causal learning and blockchain. IEEE Journal of Biomedical and Health Informatics","DOI":"10.1109\/JBHI.2024.3375894"},{"key":"2665_CR34","doi-asserted-by":"crossref","unstructured":"Han, L., Hou, J., Cho, K., Duan, R., Cai, T. (2025). Federated adaptive causal estimation (face) of target treatment effects. Journal of the American Statistical Association (just-accepted), 1\u201325","DOI":"10.1080\/01621459.2025.2453249"},{"issue":"3","key":"2665_CR35","doi-asserted-by":"publisher","first-page":"6032","DOI":"10.1109\/LRA.2021.3090020","volume":"6","author":"R Chen","year":"2021","unstructured":"Chen, R., Lu, Y., Witherell, P., Simpson, T. W., Kumara, S., & Yang, H. (2021). Ontology-driven learning of Bayesian network for causal inference and quality assurance in additive manufacturing. IEEE Robotics and Automation Letters, 6(3), 6032\u20136038.","journal-title":"IEEE Robotics and Automation Letters"},{"key":"2665_CR36","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1109\/ACCESS.2021.3139199","volume":"10","author":"EE Oliveira","year":"2021","unstructured":"Oliveira, E. E., Migu\u00e9is, V. L., & Borges, J. L. (2021). Understanding overlap in automatic root cause analysis in manufacturing using causal inference. IEEE Access, 10, 191\u2013201.","journal-title":"IEEE Access"},{"key":"2665_CR37","doi-asserted-by":"publisher","first-page":"102356","DOI":"10.1016\/j.rcim.2022.102356","volume":"77","author":"J Hua","year":"2022","unstructured":"Hua, J., Li, Y., Liu, C., & Wang, L. (2022). A zero-shot prediction method based on causal inference under non-stationary manufacturing environments for complex manufacturing systems. Robotics and Computer-Integrated Manufacturing, 77, 102356.","journal-title":"Robotics and Computer-Integrated Manufacturing"},{"issue":"13","key":"2665_CR38","doi-asserted-by":"publisher","first-page":"4717","DOI":"10.1080\/00207543.2023.2274335","volume":"62","author":"KYH Lim","year":"2024","unstructured":"Lim, K. Y. H., Yosal, T. S., Chen, C.-H., Zheng, P., Wang, L., & Xu, X. (2024). Graph-enabled cognitive digital twins for causal inference in maintenance processes. International Journal of Production Research, 62(13), 4717\u20134734.","journal-title":"Int. J. Prod. Res."},{"key":"2665_CR39","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1016\/j.jmsy.2024.07.008","volume":"76","author":"C Liu","year":"2024","unstructured":"Liu, C., Li, Y., Hua, J., Zhao, Z., & Gao, J. (2024). A causal based method for denoising non-homologous noises in time series manufacturing monitoring data. Journal of Manufacturing Systems, 76, 92\u2013102.","journal-title":"J. Manuf. Syst."},{"key":"2665_CR40","doi-asserted-by":"publisher","first-page":"110027","DOI":"10.1016\/j.engappai.2025.110027","volume":"143","author":"Q Li","year":"2025","unstructured":"Li, Q., Huang, T., Liu, J., & Wang, S. (2025). Causality-guided fault diagnosis under visual interference in fused deposition modeling. Engineering Applications of Artificial Intelligence, 143, 110027.","journal-title":"Eng. Appl. Artif. Intell."},{"issue":"518","key":"2665_CR41","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1080\/01621459.2017.1285773","volume":"112","author":"DM Blei","year":"2017","unstructured":"Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518), 859\u2013877.","journal-title":"J. Am. Stat. Assoc."},{"key":"2665_CR42","unstructured":"Minka, T.P.(2013). Expectation propagation for approximate bayesian inference. arXiv preprint arXiv:1301.2294"},{"key":"2665_CR43","unstructured":"Guo, H., Greengard, P., Wang, H., Gelman, A., Kim, Y., Xing, E.P. (2023). Federated learning as variational inference: A scalable expectation propagation approach. arXiv preprint arXiv:2302.04228"},{"issue":"1","key":"2665_CR44","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1080\/24725854.2022.2157912","volume":"56","author":"X Yue","year":"2024","unstructured":"Yue, X., Kontar, R. A., & G\u00f3mez, A. M. E. (2024). Federated data analytics: A study on linear models. IISE Transactions, 56(1), 16\u201328.","journal-title":"IISE Transactions"},{"key":"2665_CR45","unstructured":"Welling, M., Teh, Y.W.( 2011). Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th International Conference on Machine Learning (ICML-11), pp. 681\u2013 688. Citeseer"},{"issue":"17","key":"2665_CR46","first-page":"1","volume":"21","author":"A Vehtari","year":"2020","unstructured":"Vehtari, A., Gelman, A., Sivula, T., Jyl\u00e4nki, P., Tran, D., Sahai, S., Blomstedt, P., Cunningham, J. P., Schiminovich, D., & Robert, C. P. (2020). Expectation propagation as a way of life: A framework for bayesian inference on partitioned data. Journal of Machine Learning Research, 21(17), 1\u201353.","journal-title":"J. Mach. Learn. Res."},{"key":"2665_CR47","unstructured":"Li, T., Hu, S., Beirami, A., Smith, V.( 2021). Ditto: Fair and robust federated learning through personalization. In: International Conference on Machine Learning, pp. 6357\u2013 6368. PMLR"},{"key":"2665_CR48","unstructured":"Karimireddy, S.P., Kale, S., Mohri, M., Reddi, S., Stich, S., Suresh, A.T.( 2020). Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pp. 5132\u2013 5143. PMLR"},{"key":"2665_CR49","doi-asserted-by":"crossref","unstructured":"Brishty, F., Urner, R., Grau, G.(2022). Machine learning based data driven inkjet printed electronics: jetting prediction for novel inks. Flexible and Printed Electronics 7","DOI":"10.1088\/2058-8585\/ac5a39"},{"key":"2665_CR50","doi-asserted-by":"publisher","unstructured":"Lab, S.S.M.(2025) Siemens Smart Manufacturing Maintenance DS. Kaggle. https:\/\/doi.org\/10.34740\/KAGGLE\/DSV\/11400476. https:\/\/www.kaggle.com\/dsv\/11400476","DOI":"10.34740\/KAGGLE\/DSV\/11400476"}],"container-title":["Journal of Intelligent Manufacturing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10845-025-02665-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10845-025-02665-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10845-025-02665-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T04:57:43Z","timestamp":1782277063000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10845-025-02665-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,2]]},"references-count":50,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["2665"],"URL":"https:\/\/doi.org\/10.1007\/s10845-025-02665-7","relation":{},"ISSN":["0956-5515","1572-8145"],"issn-type":[{"value":"0956-5515","type":"print"},{"value":"1572-8145","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,2]]},"assertion":[{"value":"17 April 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest, financial or otherwise, related to this work.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This research did not involve human participants or animals, and thus no informed consent or ethical approval was required. All procedures and data collection complied with relevant ethical standards, and the study was conducted in accordance with institutional and national guidelines.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}]}}