{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T10:22:49Z","timestamp":1775730169847,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T00:00:00Z","timestamp":1750896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Eindhoven Artificial Intelligence Systems Institute"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We present a Forney-style factor graph representation for the class of multivariate autoregressive models with exogenous inputs, and we propose an online Bayesian parameter-identification procedure based on message passing within this graph. We derive message-update rules for (1) a custom factor node that represents the multivariate autoregressive likelihood function and (2) the matrix normal Wishart distribution over the parameters. The flow of messages reveals how parameter uncertainty propagates into predictive uncertainty over the system outputs and how individual factor nodes and edges contribute to the overall model evidence. We evaluate the message-passing-based procedure on (i) a simulated autoregressive system, demonstrating convergence, and (ii) on a benchmark task, demonstrating strong predictive performance.<\/jats:p>","DOI":"10.3390\/e27070679","type":"journal-article","created":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T06:55:13Z","timestamp":1750920913000},"page":"679","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Factor Graph-Based Online Bayesian Identification and Component Evaluation for Multivariate Autoregressive Exogenous Input Models"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-3114-812X","authenticated-orcid":false,"given":"Tim N.","family":"Nisslbeck","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0547-4817","authenticated-orcid":false,"given":"Wouter M.","family":"Kouw","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,26]]},"reference":[{"key":"ref_1","unstructured":"Nisslbeck, T.N., and Kouw, W.M. (2025, January 24\u201327). Online Bayesian system identification in multivariate autoregressive models via message passing. Proceedings of the European Control Conference, Thessaloniki, Greece. (accepted)."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1111\/j.2517-6161.1964.tb00560.x","article-title":"On the Bayesian estimation of multivariate regression","volume":"26","author":"Tiao","year":"1964","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1111\/j.2517-6161.1989.tb01759.x","article-title":"Recursive estimation of autoregressions","volume":"51","author":"Hannan","year":"1989","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1016\/B978-0-444-62731-5.00015-4","article-title":"Forecasting with Bayesian vector autoregression","volume":"2","author":"Karlsson","year":"2013","journal-title":"Handb. Econ. Forecast."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Nisslbeck, T.N., and Kouw, W.M. (2024, January 9\u201311). Coupled autoregressive active inference agents for control of multi-joint dynamical systems. Proceedings of the International Workshop on Active Inference, Oxford, UK.","DOI":"10.1007\/978-3-031-77138-5_9"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Barber, D. (2012). Bayesian Reasoning and Machine Learning, Cambridge University Press.","DOI":"10.1017\/CBO9780511804779"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1002\/jae.2751","article-title":"Mixed causal\u2013noncausal autoregressions with exogenous regressors","volume":"35","author":"Hecq","year":"2020","journal-title":"J. Appl. Econom."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Penny, W., and Harrison, L. (2007). Multivariate autoregressive models. Statistical Parametric Mapping: The Analysis of Functional Brain Images, Academic Press.","DOI":"10.1016\/B978-012372560-8\/50040-1"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1080\/03610920701504370","article-title":"Bayesian identification of multivariate autoregressive processes","volume":"37","author":"Shaarawy","year":"2008","journal-title":"Commun. Stat. Methods"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1214\/ss\/1177009939","article-title":"Bayesian experimental design: A review","volume":"10","author":"Chaloner","year":"1995","journal-title":"Stat. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1603","DOI":"10.1109\/TRO.2018.2865891","article-title":"Information-theoretic model predictive control: Theory and applications to autonomous driving","volume":"34","author":"Williams","year":"2018","journal-title":"IEEE Trans. Robot."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1109\/18.910572","article-title":"Factor graphs and the sum-product algorithm","volume":"47","author":"Kschischang","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"\u015een\u00f6z, \u0130., van de Laar, T., Bagaev, D., and de Vries, B. (2021). Variational message passing and local constraint manipulation in factor graphs. Entropy, 23.","DOI":"10.3390\/e23070807"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6314","DOI":"10.1016\/j.ifacol.2017.08.914","article-title":"Linear optimal control on factor graphs\u2014a message passing perspective","volume":"50","author":"Hoffmann","year":"2017","journal-title":"IFAC-PapersOnLine"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1109\/JPROC.2007.896497","article-title":"The factor graph approach to model-based signal processing","volume":"95","author":"Loeliger","year":"2007","journal-title":"Proc. IEEE"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.ijar.2018.11.002","article-title":"A factor graph approach to automated design of Bayesian signal processing algorithms","volume":"104","author":"Cox","year":"2019","journal-title":"Int. J. Approx. Reason."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"15193","DOI":"10.1109\/ACCESS.2022.3148127","article-title":"A unifying view of estimation and control using belief propagation with application to path planning","volume":"10","author":"Palmieri","year":"2022","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1109\/18.910573","article-title":"Codes on graphs: Normal realizations","volume":"47","author":"Forney","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Le, F., Srivatsa, M., Reddy, K.K., and Roy, K. (2019, January 4\u20137). Using graphical models as explanations in deep neural networks. Proceedings of the IEEE International Conference on Mobile Ad-Hoc and Smart Systems, Monterey, CA, USA.","DOI":"10.1109\/MASS.2019.00041"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"41","DOI":"10.3233\/SW-190374","article-title":"On the role of knowledge graphs in explainable AI","volume":"11","author":"Lecue","year":"2020","journal-title":"Semant. Web"},{"key":"ref_21","unstructured":"Yedidia, J.S., Freeman, W.T., and Weiss, Y. (2001). Bethe free energy, Kikuchi approximations, and belief propagation algorithms. Adv. Neural Inf. Process. Syst., 13, Available online: https:\/\/merl.com\/publications\/docs\/TR2001-16.pdf."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Xu, W., Liu, A., and Lau, V. (2024, January 7\u20139). Message Passing Based Wireless Federated Learning via Analog Message Aggregation. Proceedings of the IEEE\/CIC International Conference on Communications in China, Hangzhou, China.","DOI":"10.1109\/ICCC62479.2024.10682004"},{"key":"ref_23","first-page":"6601690","article-title":"Reactive message passing for scalable Bayesian inference","volume":"2023","author":"Bagaev","year":"2023","journal-title":"Sci. Program."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Podusenko, A., Kouw, W.M., and de Vries, B. (2021). Message passing-based inference for time-varying autoregressive models. Entropy, 23.","DOI":"10.3390\/e23060683"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kouw, W.M., Podusenko, A., Koudahl, M.T., and Schoukens, M. (2022, January 8\u201310). Variational message passing for online polynomial NARMAX identification. Proceedings of the American Control Conference, Atlanta, GA, USA.","DOI":"10.23919\/ACC53348.2022.9867898"},{"key":"ref_26","first-page":"510","article-title":"The matrix cookbook","volume":"7","author":"Petersen","year":"2008","journal-title":"Tech. Univ. Den."},{"key":"ref_27","unstructured":"Soch, J., Allefeld, C., Faulkenberry, T.J., Pavlovic, M., Petrykowski, K., Sar\u0131ta\u015f, K., Balkus, S., Kipnis, A., Atze, H., and Martin, O.A. (2025, June 08). The Book of Statistical Proofs (Version 2023). Available online: https:\/\/zenodo.org\/records\/10495684."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Gupta, A.K., and Nagar, D.K. (2018). Matrix Variate Distributions, Chapman and Hall\/CRC.","DOI":"10.1201\/9780203749289"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4, S. (2013). Bayesian Filtering and Smoothing, Cambridge University Press.","DOI":"10.1017\/CBO9781139344203"},{"key":"ref_30","unstructured":"Lopes, M.T., Castello, D.A., and Matt, C.F.T. (2010, January 25\u201327). A Bayesian inference approach to estimate elastic and damping parameters of a structure subjected to vibration tests. Proceedings of the Inverse Problems, Design and Optimization Symposium, Joao Pessoa, Brazil."},{"key":"ref_31","first-page":"661","article-title":"Variational message passing","volume":"6","author":"Winn","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Dauwels, J., Korl, S., and Loeliger, H.A. (2006, January 9\u201314). Particle methods as message passing. Proceedings of the IEEE International Symposium on Information Theory, Seattle, DC, USA.","DOI":"10.1109\/ISIT.2006.261910"},{"key":"ref_33","unstructured":"Murphy, K.P. (2012). Machine Learning: A Probabilistic Perspective, MIT Press."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"102632","DOI":"10.1016\/j.jmp.2021.102632","article-title":"A step-by-step tutorial on active inference and its application to empirical data","volume":"107","author":"Smith","year":"2022","journal-title":"J. Math. Psychol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1080\/01621459.2017.1285773","article-title":"Variational inference: A review for statisticians","volume":"112","author":"Blei","year":"2017","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Parr, T., Pezzulo, G., and Friston, K.J. (2022). Active Inference: The Free Energy Principle in Mind, Brain, and Behavior, MIT Press.","DOI":"10.7551\/mitpress\/12441.001.0001"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2023.07.001","article-title":"The free energy principle made simpler but not too simple","volume":"1024","author":"Friston","year":"2023","journal-title":"Phys. Rep."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2282","DOI":"10.1109\/TIT.2005.850085","article-title":"Constructing free-energy approximations and generalized belief propagation algorithms","volume":"51","author":"Yedidia","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_39","unstructured":"Proakis, J.G. (2001). Digital Signal Processing: Principles Algorithms and Applications, Pearson Education India."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/S1050-6411(03)00080-4","article-title":"Design and responses of Butterworth and critically damped digital filters","volume":"13","author":"Robertson","year":"2003","journal-title":"J. Electromyogr. Kinesiol."},{"key":"ref_41","unstructured":"Smith, J. (2008). Introduction to Digital Filters: With Audio Applications, W3K Publishing."},{"key":"ref_42","unstructured":"Zumbahlen, H. (2011). Linear Circuit Design Handbook, Newnes."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.cmpb.2007.04.004","article-title":"Digital Butterworth filter for subtracting noise from low magnitude surface electromyogram","volume":"87","author":"Mello","year":"2007","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Damgaard, M.R., Pedersen, R., and Bak, T. (2022). Study of variational inference for flexible distributed probabilistic robotics. Robotics, 11.","DOI":"10.20944\/preprints202202.0053.v1"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1666","DOI":"10.1109\/TCCN.2023.3307953","article-title":"Message passing neural network versus message passing algorithm for cooperative positioning","volume":"9","author":"Tedeschini","year":"2023","journal-title":"IEEE Trans. Cogn. Commun. Netw."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Ta, D.N., Kobilarov, M., and Dellaert, F. (2014, January 27\u201330). A factor graph approach to estimation and model predictive control on unmanned aerial vehicles. Proceedings of the International Conference on Unmanned Aircraft Systems, Orlando, FL, USA.","DOI":"10.1109\/ICUAS.2014.6842254"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Castaldo, F., and Palmieri, F.A. (2014, January 23\u201325). A multi-camera multi-target tracker based on factor graphs. Proceedings of the IEEE International Symposium on Innovations in Intelligent Systems and Applications, Alberobello, Italy.","DOI":"10.1109\/INISTA.2014.6873609"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"van Erp, B., Bagaev, D., Podusenko, A., \u015een\u00f6z, \u0130., and de Vries, B. (2024, January 10\u201312). Multi-agent trajectory planning with NUV priors. Proceedings of the American Control Conference, Toronto, ON, Canada.","DOI":"10.23919\/ACC60939.2024.10645034"},{"key":"ref_49","unstructured":"Assimakis, N., Adam, M., and Douladiris, A. (2012, January 26\u201328). Information filter and Kalman filter comparison: Selection of the faster filter. Proceedings of the Information Engineering, Chongqing, China."},{"key":"ref_50","unstructured":"Cover, T.M. (1999). Elements of Information Theory, John Wiley & Sons."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/679\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:58:58Z","timestamp":1760032738000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/679"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,26]]},"references-count":50,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["e27070679"],"URL":"https:\/\/doi.org\/10.3390\/e27070679","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,26]]}}}