{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T20:11:52Z","timestamp":1779394312392,"version":"3.53.1"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T00:00:00Z","timestamp":1779321600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T00:00:00Z","timestamp":1779321600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC 2075 - 390740016"],"award-info":[{"award-number":["EXC 2075 - 390740016"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["520388526"],"award-info":[{"award-number":["520388526"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC 2075 - 390740016"],"award-info":[{"award-number":["EXC 2075 - 390740016"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009534","name":"Universit\u00e4t Stuttgart","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009534","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2026,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Surrogate models are often used as computationally efficient approximations to complex simulation models, enabling tasks such as solving inverse problems, sensitivity analysis, and probabilistic forward predictions, which would otherwise be computationally infeasible. During training, surrogate parameters are fitted such that the surrogate reproduces the simulation model\u2019s outputs as closely as possible. However, the simulation model itself is merely a simplification of the real-world system, often missing relevant processes or suffering from misspecifications e.g., in inputs or boundary conditions. Hints about these might be captured in real-world measurement data, and yet, we typically ignore those hints during surrogate building. In this paper, we propose two novel probabilistic approaches to integrate simulation data and real-world measurement data during surrogate training. The first method trains separate surrogate models for each data source and combines their predictive distributions, while the second incorporates both data sources by training a single surrogate. Both hybrid modeling approaches employ a novel weighting strategy for combining heterogeneous data sources during surrogate training, which operates independently of the chosen surrogate family. We show the conceptual differences and benefits of the two approaches through both synthetic and real-world case studies. The results demonstrate the potential of these methods to improve predictive accuracy, predictive coverage, and to diagnose problems in the underlying simulation model. These insights can improve system understanding and future model development.<\/jats:p>","DOI":"10.1007\/s11222-026-10906-9","type":"journal-article","created":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T19:05:24Z","timestamp":1779390324000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bayesian surrogate training on multiple data sources: a hybrid modeling strategy"],"prefix":"10.1007","volume":"36","author":[{"given":"Philipp","family":"Reiser","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paul-Christian","family":"B\u00fcrkner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anneli","family":"Guthke","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,21]]},"reference":[{"key":"10906_CR1","doi-asserted-by":"publisher","DOI":"10.7717\/peerj-cs.1516","volume":"9","author":"O Abril-Pla","year":"2023","unstructured":"Abril-Pla, O., Andr\u00e9ani, V., Carroll, C., Dong, L., Fonnesbeck, C., Kochurov, M., Kumar, R., Lao, J., Luhmann, C.C., Martin, O.A., Osthege, M., Vieira, R., Wiecki, T.V., Zinkov, R.: PyMC: a modern, and comprehensive probabilistic programming framework in Python. PeerJ Comput. Sci. 9, e1516 (2023)","journal-title":"PeerJ Comput. Sci."},{"issue":"1","key":"10906_CR2","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/s00180-011-0301-1","volume":"28","author":"C Agostinelli","year":"2013","unstructured":"Agostinelli, C., Greco, L.: A weighted strategy to handle likelihood uncertainty in Bayesian inference. Comput. Statistics 28(1), 319\u2013339 (2013)","journal-title":"Comput. Statistics"},{"issue":"1","key":"10906_CR3","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1109\/TCBB.2018.2843339","volume":"17","author":"K Alden","year":"2020","unstructured":"Alden, K., Cosgrove, J., Coles, M., Timmis, J.: Using Emulation to Engineer and Understand Simulations of Biological Systems. IEEE\/ACM Trans. Comput. Biol. Bioinf. 17(1), 302\u2013315 (2020)","journal-title":"IEEE\/ACM Trans. Comput. Biol. Bioinf."},{"issue":"1","key":"10906_CR4","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1214\/09-BA404","volume":"4","author":"MJ Bayarri","year":"2009","unstructured":"Bayarri, M.J., Berger, J.O., Liu, F.: Modularization in Bayesian analysis, with emphasis on analysis of computer models. Bayesian Anal. 4(1), 119\u2013150 (2009)","journal-title":"Bayesian Anal."},{"issue":"5","key":"10906_CR5","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1111\/rssb.12158","volume":"78","author":"PG Bissiri","year":"2016","unstructured":"Bissiri, P.G., Holmes, C., Walker, S.: A General Framework for Updating Belief Distributions. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 78(5), 1103\u20131130 (2016)","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"10906_CR6","unstructured":"Bosch, N., Hennig, P., Tronarp, F.: Calibrated Adaptive Probabilistic ODE Solvers. In Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, pages 3466\u20133474. PMLR (2021)"},{"key":"10906_CR7","unstructured":"Bridgman, W., Balakrishnan, U., Jones, R., Chen, J., Wu, X., Safta, C., Huang, Y., Khalil, M.: Enhancing Polynomial Chaos Expansion Based Surrogate Modeling using a Novel Probabilistic Transfer Learning. Strategy (2023)"},{"key":"10906_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112210","volume":"488","author":"P-C B\u00fcrkner","year":"2023","unstructured":"B\u00fcrkner, P.-C., Kr\u00f6ker, I., Oladyshkin, S., Nowak, W.: A fully Bayesian sparse polynomial chaos expansion approach with joint priors on the coefficients and global selection of terms. J. Comput. Phys. 488, 112210 (2023)","journal-title":"J. Comput. Phys."},{"key":"10906_CR9","unstructured":"Carmona, C.U., Nicholls, G.K.: Semi-modular inference: enhanced learning in multi-modular models by tempering the influence of components. In Chiappa, S. and Calandra, R., editors, Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics, AISTATS 2020, volume 108 of Proceedings of Machine Learning Research, pages 4226\u20134235. PMLR (2020)"},{"issue":"1","key":"10906_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v076.i01","volume":"76","author":"B Carpenter","year":"2017","unstructured":"Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., Riddell, A.: Stan: A probabilistic programming language. J. Stat. Softw. 76(1), 1\u201332 (2017)","journal-title":"J. Stat. Softw."},{"key":"10906_CR11","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1016\/j.enbuild.2017.08.069","volume":"154","author":"A Chong","year":"2017","unstructured":"Chong, A., Lam, K.P., Pozzi, M., Yang, J.: Bayesian calibration of building energy models with large datasets. Energy and Buildings 154, 343\u2013355 (2017)","journal-title":"Energy and Buildings"},{"key":"10906_CR12","doi-asserted-by":"publisher","first-page":"123149","DOI":"10.1109\/ACCESS.2023.3329685","volume":"11","author":"F Fiedler","year":"2023","unstructured":"Fiedler, F., Lucia, S.: Improved Uncertainty Quantification for Neural Networks With Bayesian Last Layer. IEEE Access 11, 123149\u2013123160 (2023)","journal-title":"IEEE Access"},{"key":"10906_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41591-020-0883-7","volume":"26","author":"G Giordano","year":"2020","unstructured":"Giordano, G., Blanchini, F., Bruno, R., Colaneri, P., Filippo, A., Di Matteo, A., Colaneri, M.: Modelling the covid-19 epidemic and implementation of population-wide interventions in italy. Nat. Med. 26, 1\u20136 (2020)","journal-title":"Nat. Med."},{"key":"10906_CR14","unstructured":"Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press. (2016). http:\/\/www.deeplearningbook.org"},{"key":"10906_CR15","doi-asserted-by":"crossref","unstructured":"Gramacy, R.B.: Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences. Chapman Hall\/CRC, Boca Raton, Florida. (2020). http:\/\/bobby.gramacy.com\/surrogates\/","DOI":"10.1201\/9780367815493"},{"issue":"4","key":"10906_CR16","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1214\/17-BA1085","volume":"12","author":"P Gr\u00fcnwald","year":"2017","unstructured":"Gr\u00fcnwald, P., van Ommen, T.: Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It. Bayesian Anal. 12(4), 1069\u20131103 (2017)","journal-title":"Bayesian Anal."},{"issue":"1","key":"10906_CR17","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1038\/s41597-022-01245-1","volume":"9","author":"E Guidotti","year":"2022","unstructured":"Guidotti, E.: A worldwide epidemiological database for covid-19 at fine-grained spatial resolution. Scientific Data 9(1), 112 (2022)","journal-title":"Scientific Data"},{"issue":"51","key":"10906_CR18","doi-asserted-by":"publisher","first-page":"2376","DOI":"10.21105\/joss.02376","volume":"5","author":"E Guidotti","year":"2020","unstructured":"Guidotti, E., Ardia, D.: Covid-19 data hub. Journal of Open Source Software 5(51), 2376 (2020)","journal-title":"Journal of Open Source Software"},{"key":"10906_CR19","unstructured":"Harrison, J., Willes, J., Snoek, J.: Variational Bayesian Last Layers. In: The Twelfth International Conference on Learning Representations, (2023)"},{"issue":"4","key":"10906_CR20","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1137\/S0036144500371907","volume":"42","author":"HW Hethcote","year":"2000","unstructured":"Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599\u2013653 (2000)","journal-title":"SIAM Rev."},{"key":"10906_CR21","unstructured":"Holmes, C., Walker, S.: Assigning a value to a power likelihood in a general Bayesian model, (2017)"},{"issue":"1","key":"10906_CR22","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s11222-023-10366-5","volume":"34","author":"N Kallioinen","year":"2024","unstructured":"Kallioinen, N., Paananen, T., B\u00fcrkner, P.-C., Vehtari, A.: Detecting and diagnosing prior and likelihood sensitivity with power-scaling. Stat. Comput. 34(1), 57 (2024)","journal-title":"Stat. Comput."},{"issue":"3","key":"10906_CR23","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1111\/1467-9868.00294","volume":"63","author":"MC Kennedy","year":"2001","unstructured":"Kennedy, M.C., O\u2019Hagan, A.: Bayesian calibration of computer models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63(3), 425\u2013464 (2001)","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"10906_CR24","unstructured":"Kristiadi, A., Hein, M., Hennig, P.: Being bayesian, even just a bit, fixes overconfidence in relu networks. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pages 5436\u20135446. PMLR (2020)"},{"issue":"3","key":"10906_CR25","doi-asserted-by":"publisher","first-page":"A1163","DOI":"10.1137\/130938189","volume":"36","author":"J Li","year":"2014","unstructured":"Li, J., Marzouk, Y.M.: Adaptive construction of surrogates for the bayesian solution of inverse problems. SIAM J. Sci. Comput. 36(3), A1163\u2013A1186 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"10906_CR26","doi-asserted-by":"crossref","unstructured":"McLatchie, Y., Fong, E., Frazier, D.T., Knoblauch, J.: Predictive performance of power posteriors (2024)","DOI":"10.1093\/biomet\/asaf034"},{"key":"10906_CR27","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/j.advwatres.2018.05.007","volume":"117","author":"F Mohammadi","year":"2018","unstructured":"Mohammadi, F., Kopmann, R., Guthke, A., Oladyshkin, S., Nowak, W.: Bayesian selection of hydro-morphodynamic models under computational time constraints. Adv. Water Resour. 117, 53\u201364 (2018)","journal-title":"Adv. Water Resour."},{"key":"10906_CR28","doi-asserted-by":"crossref","unstructured":"Neal, R.M.: Bayesian learning for neural networks, University of Toronto, Canada (1995). (PhD thesis)","DOI":"10.1007\/978-1-4612-0745-0"},{"key":"10906_CR29","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/j.ress.2012.05.002","volume":"106","author":"S Oladyshkin","year":"2012","unstructured":"Oladyshkin, S., Nowak, W.: Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliability Engineering & System Safety 106, 179\u2013190 (2012)","journal-title":"Reliability Engineering & System Safety"},{"issue":"2","key":"10906_CR30","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1007\/s11222-020-09982-2","volume":"31","author":"T Paananen","year":"2021","unstructured":"Paananen, T., Piironen, J., B\u00fcrkner, P.-C., Vehtari, A.: Implicitly adaptive importance sampling. Stat. Comput. 31(2), 16 (2021)","journal-title":"Stat. Comput."},{"issue":"1","key":"10906_CR31","first-page":"6","volume":"3","author":"S Ranftl","year":"2021","unstructured":"Ranftl, S., von der Linden, W.: Bayesian Surrogate Analysis and Uncertainty Propagation. Physical Sciences Forum 3(1), 6 (2021)","journal-title":"Physical Sciences Forum"},{"key":"10906_CR32","unstructured":"Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge, Mass., 3. print edition (2008)"},{"issue":"3","key":"10906_CR33","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1007\/s11222-025-10597-8","volume":"35","author":"P Reiser","year":"2025","unstructured":"Reiser, P., Aguilar, J.E., Guthke, A., B\u00fcrkner, P.-C.: Uncertainty quantification and propagation in surrogate-based bayesian inference. Stat. Comput. 35(3), 66 (2025)","journal-title":"Stat. Comput."},{"issue":"5","key":"10906_CR34","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1006181","volume":"14","author":"M Renardy","year":"2018","unstructured":"Renardy, M., Yi, T.-M., Xiu, D., Chou, C.-S.: Parameter uncertainty quantification using surrogate models applied to a spatial model of yeast mating polarization. PLoS Comput. Biol. 14(5), e1006181 (2018)","journal-title":"PLoS Comput. Biol."},{"key":"10906_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4145-2","volume-title":"Monte Carlo statistical methods","author":"C Robert","year":"2004","unstructured":"Robert, C., Casella, G.: Monte Carlo statistical methods. Springer Verlag (2004)"},{"issue":"2","key":"10906_CR36","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1111\/1467-9868.00392","volume":"65","author":"R Royall","year":"2003","unstructured":"Royall, R., Tsou, T.-S.: Interpreting Statistical Evidence by using Imperfect Models: Robust Adjusted Likelihood Functions. J. R. Stat. Soc. Ser. B Stat Methodol. 65(2), 391\u2013404 (2003)","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"key":"10906_CR37","doi-asserted-by":"crossref","unstructured":"Scheurer, S., Sch\u00e4fer Rodrigues Silva, A., Mohammadi, F., Hommel, J., Oladyshkin, S., Flemisch, B., Nowak, W.: Surrogate-based Bayesian comparison of computationally expensive models: Application to microbially induced calcite precipitation. Comput. Geosci. 25(6), 1899\u20131917 (2021)","DOI":"10.1007\/s10596-021-10076-9"},{"key":"10906_CR38","unstructured":"Schmidt, J., Kr\u00e4mer, N., Hennig, P.: A Probabilistic State Space Model for Joint Inference from Differential Equations and Data. In Advances in Neural Information Processing Systems, volume 34, pages 12374\u201312385. Curran Associates, Inc (2021)"},{"key":"10906_CR39","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1016\/j.cma.2017.01.033","volume":"318","author":"Q Shao","year":"2017","unstructured":"Shao, Q., Younes, A., Fahs, M., Mara, T.A.: Bayesian sparse polynomial chaos expansion for global sensitivity analysis. Comput. Methods Appl. Mech. Eng. 318, 474\u2013496 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"4","key":"10906_CR40","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/0041-5553(67)90144-9","volume":"7","author":"IM Sobol\u2019","year":"1967","unstructured":"Sobol\u2019, I.M.: On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7(4), 86\u2013112 (1967)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"10906_CR41","first-page":"35","volume":"2","author":"Stan Development Team","year":"2024","unstructured":"Stan Development Team: Stan Modeling Language Users Guide and Reference Manual. Version 2, 35 (2024)","journal-title":"Version"},{"issue":"7","key":"10906_CR42","doi-asserted-by":"publisher","first-page":"964","DOI":"10.1016\/j.ress.2007.04.002","volume":"93","author":"B Sudret","year":"2008","unstructured":"Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety 93(7), 964\u2013979 (2008)","journal-title":"Reliability Engineering & System Safety"},{"key":"10906_CR43","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4, S., Svensson, L.: Bayesian Filtering and Smoothing. In: Institute of Mathematical Statistics Textbooks, Cambridge University Press (2023) . (2 edition)","DOI":"10.1017\/9781108917407"},{"key":"10906_CR44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.108909","volume":"399","author":"A Tarakanov","year":"2019","unstructured":"Tarakanov, A., Elsheikh, A.H.: Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models. J. Comput. Phys. 399, 108909 (2019)","journal-title":"J. Comput. Phys."},{"key":"10906_CR45","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1016\/j.jcp.2019.03.039","volume":"388","author":"E Torre","year":"2019","unstructured":"Torre, E., Marelli, S., Embrechts, P., Sudret, B.: Data-driven polynomial chaos expansion for machine learning regression. J. Comput. Phys. 388, 601\u2013623 (2019)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"10906_CR46","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1007\/s11222-019-09900-1","volume":"29","author":"F Tronarp","year":"2019","unstructured":"Tronarp, F., Kersting, H., S\u00e4rkk\u00e4, S., Hennig, P.: Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: A new perspective. Stat. Comput. 29(6), 1297\u20131315 (2019)","journal-title":"Stat. Comput."},{"issue":"5","key":"10906_CR47","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1007\/s11222-016-9696-4","volume":"27","author":"A Vehtari","year":"2017","unstructured":"Vehtari, A., Gelman, A., Gabry, J.: Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Stat. Comput. 27(5), 1413\u20131432 (2017)","journal-title":"Stat. Comput."},{"key":"10906_CR48","doi-asserted-by":"crossref","unstructured":"Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., B\u00fcrkner, P.-C.: Rank-normalization, folding, and localization: An improved $$\\widehat{R}$$ for assessing convergence of MCMC. Bayesian Anal. 16(2), (2021)","DOI":"10.1214\/20-BA1221"},{"key":"10906_CR49","doi-asserted-by":"crossref","unstructured":"Vehtari, A., Ojanen, J.: A survey of Bayesian predictive methods for model assessment, selection and comparison. Statistics Surveys, 6(none):142\u2013228 (2012)","DOI":"10.1214\/12-SS102"},{"issue":"4","key":"10906_CR50","doi-asserted-by":"publisher","first-page":"897","DOI":"10.2307\/2371268","volume":"60","author":"N Wiener","year":"1938","unstructured":"Wiener, N.: The Homogeneous Chaos. Am. J. Math. 60(4), 897 (1938)","journal-title":"Am. J. Math."},{"issue":"3","key":"10906_CR51","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1214\/17-BA1091","volume":"13","author":"Y Yao","year":"2018","unstructured":"Yao, Y., Vehtari, A., Simpson, D., Gelman, A.: Using Stacking to Average Bayesian Predictive Distributions (with Discussion). Bayesian Anal. 13(3), 917\u20131007 (2018)","journal-title":"Bayesian Anal."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-026-10906-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-026-10906-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-026-10906-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T19:05:31Z","timestamp":1779390331000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-026-10906-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,21]]},"references-count":51,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["10906"],"URL":"https:\/\/doi.org\/10.1007\/s11222-026-10906-9","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,21]]},"assertion":[{"value":"16 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 May 2026","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 competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"145"}}