{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:15:24Z","timestamp":1775693724921,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T00:00:00Z","timestamp":1670976000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T00:00:00Z","timestamp":1670976000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002341","name":"Academy of Finland","doi-asserted-by":"crossref","award":["298742"],"award-info":[{"award-number":["298742"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland","doi-asserted-by":"crossref","award":["308640"],"award-info":[{"award-number":["308640"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland","doi-asserted-by":"crossref","award":["313122"],"award-info":[{"award-number":["313122"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004587","name":"Instituto de Salud Carlos III","doi-asserted-by":"publisher","award":["CD21\/00186"],"award-info":[{"award-number":["CD21\/00186"]}],"id":[{"id":"10.13039\/501100004587","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Finnish Center for Artificial Intelligence, and Technology Industries of Finland Centennial Foundation","award":["70007503"],"award-info":[{"award-number":["70007503"]}]},{"DOI":"10.13039\/501100016386","name":"Conselleria de Innovaci\u00f3n, Universidades, Ciencia y Sociedad Digital, Generalitat Valenciana","doi-asserted-by":"publisher","award":["AICO\/2020\/285"],"award-info":[{"award-number":["AICO\/2020\/285"]}],"id":[{"id":"10.13039\/501100016386","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Gaussian processes are powerful non-parametric probabilistic models for stochastic functions. However, the direct implementation entails a complexity that is computationally intractable when the number of observations is large, especially when estimated with fully Bayesian methods such as Markov chain Monte Carlo. In this paper, we focus on a low-rank approximate Bayesian Gaussian processes, based on a basis function approximation via Laplace eigenfunctions for stationary covariance functions. The main contribution of this paper is a detailed analysis of the performance, and practical recommendations for how to select the number of basis functions and the boundary factor. Intuitive visualizations and recommendations, make it easier for users to improve approximation accuracy and computational performance. We also propose diagnostics for checking that the number of basis functions and the boundary factor are adequate given the data. The approach is simple and exhibits an attractive computational complexity due to its linear structure, and it is easy to implement in probabilistic programming frameworks. Several illustrative examples of the performance and applicability of the method in the probabilistic programming language Stan are presented together with the underlying Stan model code.<\/jats:p>","DOI":"10.1007\/s11222-022-10167-2","type":"journal-article","created":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T02:02:42Z","timestamp":1670983362000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":43,"title":["Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming"],"prefix":"10.1007","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7645-4194","authenticated-orcid":false,"given":"Gabriel","family":"Riutort-Mayol","sequence":"first","affiliation":[]},{"given":"Paul-Christian","family":"B\u00fcrkner","sequence":"additional","affiliation":[]},{"given":"Michael R.","family":"Andersen","sequence":"additional","affiliation":[]},{"given":"Arno","family":"Solin","sequence":"additional","affiliation":[]},{"given":"Aki","family":"Vehtari","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,12,14]]},"reference":[{"key":"10167_CR1","volume-title":"Handbook of Mathematical Functions","author":"M Abramowitz","year":"1970","unstructured":"Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover Publishing, New York (1970)"},{"key":"10167_CR2","volume-title":"The Geometry of Random Fields","author":"RJ Adler","year":"1981","unstructured":"Adler, R.J.: The Geometry of Random Fields. SIAM, Philadelphia (1981)"},{"key":"10167_CR3","volume-title":"Theory of Linear Operators in Hilbert Space","author":"NI Akhiezer","year":"1993","unstructured":"Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover, New York (1993)"},{"issue":"139","key":"10167_CR4","first-page":"1","volume":"18","author":"MR Andersen","year":"2017","unstructured":"Andersen, M.R., Vehtari, A., Winther, O., Hansen, L.K.: Bayesian inference for spatio-temporal spike-and-slab priors. J. Mach. Learn. Res. 18(139), 1\u201358 (2017)","journal-title":"J. Mach. Learn. Res."},{"key":"10167_CR5","unstructured":"Betancourt, M.: A conceptual introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434 (2017)"},{"key":"10167_CR6","doi-asserted-by":"crossref","unstructured":"Betancourt, M., Girolami, M.: Hamiltonian Monte Carlo for hierarchical models. In: Current Trends in Bayesian Methodology with Applications. Chapman and Hall\/CRC, pp. 79\u2013101 (2019)","DOI":"10.1201\/b18502-5"},{"key":"10167_CR7","unstructured":"Briol, F.X., Oates, C., Girolami, M., Osborne, M.A., Sejdinovic, D.: Probabilistic integration: a role in statistical computation? arXiv preprint arXiv:1512.00933 (2015)"},{"key":"10167_CR8","doi-asserted-by":"publisher","DOI":"10.1201\/b10905","volume-title":"Handbook of Markov Chain Monte Carlo","author":"S Brooks","year":"2011","unstructured":"Brooks, S., Gelman, A., Jones, G., Meng, X.L.: Handbook of Markov Chain Monte Carlo. CRC Press, London (2011)"},{"issue":"1","key":"10167_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v080.i01","volume":"80","author":"PC B\u00fcrkner","year":"2017","unstructured":"B\u00fcrkner, P.C.: brms: an R package for Bayesian multilevel models using Stan. J. Stat. Softw. 80(1), 1\u201328 (2017)","journal-title":"J. Stat. Softw."},{"key":"10167_CR10","unstructured":"Burt, D., Rasmussen, C.E., Van Der Wilk, M.: Rates of convergence for sparse variational Gaussian process regression. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning, PMLR, Proceedings of Machine Learning Research, vol. 97, pp. 862\u2013871 (2019)"},{"key":"10167_CR11","volume-title":"Hierarchical Modeling and Analysis for Spatial Data","author":"BP Carlin","year":"2014","unstructured":"Carlin, B.P., Gelfand, A.E., Banerjee, S.: Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall\/CRC, London (2014)"},{"issue":"1","key":"10167_CR12","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":"10167_CR13","volume-title":"Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications","author":"H Cram\u00e9r","year":"2013","unstructured":"Cram\u00e9r, H., Leadbetter, M.R.: Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications. Courier Corporation, North Chelmsford (2013)"},{"key":"10167_CR14","unstructured":"Csat\u00f3, L., Fokou\u00e9, E., Opper, M., Schottky, B., Winther, O.: Efficient approaches to Gaussian process classification. In: Advances in Neural Information Processing Systems, pp. 251\u2013257 (2000)"},{"issue":"2","key":"10167_CR15","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1109\/TPAMI.2013.218","volume":"37","author":"MP Deisenroth","year":"2015","unstructured":"Deisenroth, M.P., Fox, D., Rasmussen, C.E.: Gaussian processes for data-efficient learning in robotics and control. IEEE Trans. Pattern Anal. Mach. Intell. 37(2), 408\u2013423 (2015)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"10167_CR16","doi-asserted-by":"publisher","DOI":"10.1201\/b15326","volume-title":"Statistical Analysis of Spatial and Spatio-temporal Point Patterns","author":"PJ Diggle","year":"2013","unstructured":"Diggle, P.J.: Statistical Analysis of Spatial and Spatio-temporal Point Patterns. Chapman and Hall\/CRC, London (2013)"},{"issue":"1","key":"10167_CR17","first-page":"57","volume":"36","author":"EM Furrer","year":"2007","unstructured":"Furrer, E.M., Nychka, D.W.: A framework to understand the asymptotic properties of kriging and splines. J. Korean Stat. Soc. 36(1), 57\u201376 (2007)","journal-title":"J. Korean Stat. Soc."},{"key":"10167_CR18","unstructured":"Gal, Y., Turner, R.: Improving the Gaussian process sparse spectrum approximation by representing uncertainty in frequency inputs. In: Bach, F., Blei, D. (eds.) Proceedings of the 32nd International Conference on Machine Learning, PMLR, Proceedings of Machine Learning Research, vol. 37, pp. 655\u2013664 (2015)"},{"key":"10167_CR19","doi-asserted-by":"publisher","DOI":"10.1201\/b16018","volume-title":"Bayesian Data Analysis","author":"A Gelman","year":"2013","unstructured":"Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian Data Analysis. Chapman and Hall\/CRC, London (2013)"},{"key":"10167_CR20","doi-asserted-by":"publisher","DOI":"10.1017\/9781139161879","volume-title":"Regression and Other Stories","author":"A Gelman","year":"2020","unstructured":"Gelman, A., Hill, J., Vehtari, A.: Regression and Other Stories. Cambridge University Press, Cambridge (2020)"},{"issue":"6","key":"10167_CR21","doi-asserted-by":"publisher","first-page":"1458","DOI":"10.1109\/72.883477","volume":"11","author":"MN Gibbs","year":"2000","unstructured":"Gibbs, M.N., MacKay, D.J.: Variational Gaussian process classifiers. IEEE Trans. Neural Netw. 11(6), 1458\u20131464 (2000)","journal-title":"IEEE Trans. Neural Netw."},{"key":"10167_CR22","unstructured":"GPy: GPy: a Gaussian process framework in Python. http:\/\/github.com\/SheffieldML\/GPy (2012)"},{"key":"10167_CR23","volume-title":"Abstract Inference","author":"U Grenander","year":"1981","unstructured":"Grenander, U.: Abstract Inference. Wiley, Hoboken, NJ (1981)"},{"issue":"2179","key":"10167_CR24","doi-asserted-by":"publisher","first-page":"20150142","DOI":"10.1098\/rspa.2015.0142","volume":"471","author":"P Hennig","year":"2015","unstructured":"Hennig, P., Osborne, M.A., Girolami, M.: Probabilistic numerics and uncertainty in computations. Proc. R. Soc. A: Math. Phys. Eng. Sci. 471(2179), 20150142 (2015)","journal-title":"Proc. R. Soc. A: Math. Phys. Eng. Sci."},{"issue":"1","key":"10167_CR25","first-page":"5537","volume":"18","author":"J Hensman","year":"2017","unstructured":"Hensman, J., Durrande, N., Solin, A.: Variational Fourier features for Gaussian processes. J. Mach. Learn. Res. 18(1), 5537\u20135588 (2017)","journal-title":"J. Mach. Learn. Res."},{"issue":"10","key":"10167_CR26","first-page":"1","volume":"90","author":"S Jo","year":"2019","unstructured":"Jo, S., Choi, T., Park, B., Lenk, P.: bsamGP: an R package for Bayesian spectral analysis models using Gaussian process priors. J. Stat. Softw. Artic. 90(10), 1\u201341 (2019)","journal-title":"J. Stat. Softw. Artic."},{"key":"10167_CR27","unstructured":"L\u00e1zaro Gredilla, M.: Sparse Gaussian processes for large-scale machine learning. Ph.D. thesis, Universidad Carlos III de Madrid (2010)"},{"key":"10167_CR28","doi-asserted-by":"publisher","first-page":"100599","DOI":"10.1016\/j.spasta.2022.100599","volume":"50","author":"F Lindgren","year":"2022","unstructured":"Lindgren, F., Bolin, D., Rue, H.: The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running. Spatial Stat. 50, 100599 (2022)","journal-title":"Spatial Stat."},{"key":"10167_CR29","volume-title":"Probability Theory","author":"M Lo\u00e8ve","year":"1977","unstructured":"Lo\u00e8ve, M.: Probability Theory. Springer-Verlag, New York (1977)"},{"issue":"40","key":"10167_CR30","first-page":"1","volume":"18","author":"AGG Matthews","year":"2017","unstructured":"Matthews, A.G.G., van der Wilk, M., Nickson, T., Fujii, K., Boukouvalas, A., Le\u00f3n-Villagr\u00e1, P., Ghahramani, Z., Hensman, J.: GPflow: a Gaussian process library using TensorFlow. J. Mach. Learn. Res. 18(40), 1\u20136 (2017)","journal-title":"J. Mach. Learn. Res."},{"key":"10167_CR31","unstructured":"Minka, T.P.: Expectation propagation for approximate Bayesian inference. In: Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann Publishers Inc., pp. 362\u2013369 (2001)"},{"key":"10167_CR32","unstructured":"Neal, R.M.: Monte Carlo implementation of Gaussian process models for Bayesian regression and classification. arXiv preprint physics\/9701026 (1997)"},{"issue":"Dec","key":"10167_CR33","first-page":"1939","volume":"6","author":"J Qui\u00f1onero-Candela","year":"2005","unstructured":"Qui\u00f1onero-Candela, J., Rasmussen, C.E.: A unifying view of sparse approximate Gaussian process regression. J. Mach. Learn. Res. 6(Dec), 1939\u20131959 (2005)","journal-title":"J. Mach. Learn. Res."},{"issue":"Jun","key":"10167_CR34","first-page":"1865","volume":"11","author":"J Qui\u00f1onero-Candela","year":"2010","unstructured":"Qui\u00f1onero-Candela, J., Rasmussen, C.E., Figueiras-Vidal, A.R.: Sparse spectrum Gaussian process regression. J. Mach. Learn. Res. 11(Jun), 1865\u20131881 (2010)","journal-title":"J. Mach. Learn. Res."},{"key":"10167_CR35","unstructured":"R Core Team: R: a language and environment for statistical computing. http:\/\/www.R-project.org\/ (2019)"},{"key":"10167_CR36","first-page":"1177","volume-title":"Advances in Neural Information Processing Systems","author":"A Rahimi","year":"2008","unstructured":"Rahimi, A., Recht, B.: Random features for large-scale kernel machines. In: Platt, J.C., Koller, D., Singer, Y., Roweis, S.T. (eds.) Advances in Neural Information Processing Systems, vol. 20, pp. 1177\u20131184. Curran Associates Inc., Red Hook (2008)"},{"key":"10167_CR37","first-page":"1313","volume-title":"Advances in Neural Information Processing Systems","author":"A Rahimi","year":"2009","unstructured":"Rahimi, A., Recht, B.: Weighted sums of random kitchen sinks: replacing minimization with randomization in learning. In: Koller, D., Schuurmans, D., Bengio, Y., Bottou, L. (eds.) Advances in Neural Information Processing Systems, vol. 21, pp. 1313\u20131320. Curran Associates Inc, Red Hook (2009)"},{"key":"10167_CR38","first-page":"3011","volume":"11","author":"CE Rasmussen","year":"2010","unstructured":"Rasmussen, C.E., Nickisch, H.: Gaussian processes for machine learning (GPML) toolbox. J. Mach. Learn. Res. 11, 3011\u20133015 (2010)","journal-title":"J. Mach. Learn. Res."},{"key":"10167_CR39","volume-title":"Gaussian Processes for Machine Learning","author":"CE Rasmussen","year":"2006","unstructured":"Rasmussen, C.E., Williams, C.K.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)"},{"key":"10167_CR40","unstructured":"Roberts, S.J.: Bayesian Gaussian processes for sequential prediction, optimisation and quadrature. Ph.D. thesis, University of Oxford (2010)"},{"key":"10167_CR41","unstructured":"Solin, A., S\u00e4rkk\u00e4, S.: Explicit link between periodic covariance functions and state space models. In: Kaski, S., Corander, J. (eds.) Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, PMLR, Proceedings of Machine Learning Research, vol. 33, pp. 904\u2013912 (2014)"},{"key":"10167_CR42","doi-asserted-by":"crossref","unstructured":"Solin, A., S\u00e4rkk\u00e4, S.: Hilbert space methods for reduced-rank Gaussian process regression. Stat. Comput. 30(2), 419\u2013446 (2020). much of the work in this paper is based on the pre-print version predating the published paper. Pre-print available at https:\/\/arxiv.org\/abs\/1401.5508","DOI":"10.1007\/s11222-019-09886-w"},{"issue":"4","key":"10167_CR43","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1109\/MSP.2013.2246292","volume":"30","author":"S S\u00e4rkk\u00e4","year":"2013","unstructured":"S\u00e4rkk\u00e4, S., Solin, A., Hartikainen, J.: Spatiotemporal learning via infinite-dimensional Bayesian filtering and smoothing: a look at Gaussian process regression through Kalman filtering. IEEE Signal Process. Mag. 30(4), 51\u201361 (2013)","journal-title":"IEEE Signal Process. Mag."},{"key":"10167_CR44","unstructured":"Stan Development Team: Stan modeling language users guide and reference manual, 2.28. https:\/\/mc-stan.org (2021)"},{"key":"10167_CR45","unstructured":"Van Trees, H.L.: Detection, Estimation, and Modulation Theory, Part I: Detection, Estimation, and Linear Modulation Theory. John Wiley & Sons, New York, NY (1968)"},{"key":"10167_CR46","unstructured":"Vanhatalo, J., Riihim\u00e4ki, J., Hartikainen, J., Jyl\u00e4nki, P., Tolvanen, V., Vehtari, A.: GPstuff: Bayesian modeling with Gaussian processes. J. Mach. Learn. Res. 14(1), 1175\u20131179 (2013)"},{"key":"10167_CR47","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1214\/12-SS102","volume":"6","author":"A Vehtari","year":"2012","unstructured":"Vehtari, A., Ojanen, J.: A survey of Bayesian predictive methods for model assessment, selection and comparison. Stat. Surv. 6, 142\u2013228 (2012)","journal-title":"Stat. Surv."},{"key":"10167_CR48","doi-asserted-by":"crossref","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)","DOI":"10.1007\/s11222-016-9696-4"},{"key":"10167_CR49","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970128","volume-title":"Spline Models for Observational Data","author":"G Wahba","year":"1990","unstructured":"Wahba, G.: Spline Models for Observational Data, vol. 59. SIAM, Philadelphia (1990)"},{"issue":"12","key":"10167_CR50","doi-asserted-by":"publisher","first-page":"1342","DOI":"10.1109\/34.735807","volume":"20","author":"CK Williams","year":"1998","unstructured":"Williams, C.K., Barber, D.: Bayesian classification with Gaussian processes. IEEE Trans. Pattern Anal. Mach. Intell. 20(12), 1342\u20131351 (1998)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-022-10167-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-022-10167-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-022-10167-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,13]],"date-time":"2023-02-13T22:48:51Z","timestamp":1676328531000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-022-10167-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,14]]},"references-count":50,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["10167"],"URL":"https:\/\/doi.org\/10.1007\/s11222-022-10167-2","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,14]]},"assertion":[{"value":"14 September 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 October 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 December 2022","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 that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"17"}}