13100 10.1140/epjc/s10052-024-13100-1 716 63 Bayesian inference with Gaussian processes for the determination of parton distribution functions 01 Mathematical Sciences 0104 Statistics Regular Article - Theoretical Physics 1 17 2024 7 10 2024 5 7 2024 7 1 2024 7 22 The Author(s) 2024 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . Funded by SCOAP3. 10052 84 84 7 7 Alessandro Candido Luigi Del Debbio Tommaso Giani tgiani@nikhef.nl Giacomo Petrillo grid.9132.9 0000 0001 2156 142X Theoretical Physics Department CERN 1211 Geneva 23 Switzerland grid.4305.2 0000 0004 1936 7988 Higgs Centre for Theoretical Physics, School of Physics and Astronomy Peter Guthrie Tait Road Edinburgh EH9 3 FD UK grid.12380.38 0000 0004 1754 9227 Department of Physics and Astronomy Vrije Universiteit 1081 HV Amsterdam The Netherlands grid.420012.5 0000 0004 0646 2193 Nikhef Theory Group Science Park 105 1098 XG Amsterdam The Netherlands grid.8404.8 0000 0004 1757 2304 Dipartimento di Statistica, Informatica, Applicazioni “Giuseppe Parenti” (DISIA) Università di Firenze Viale Morgagni 59 50134 Firenze Italy Abstract We discuss a Bayesian methodology for the solution of the inverse problem underlying the determination of parton distribution functions (PDFs). In our approach, Gaussian processes (GPs) are used to model the PDF prior, while Bayes’ theorem is used in order to determine the posterior distribution of the PDFs given a set of data. We discuss the general formalism, the Bayesian inference at the level of both parameters and hyperparameters, and the simplifications which occur when the observable entering the analysis is linear in the PDF. We benchmark the new methodology in two simple examples for the determination of a single PDF flavor from a set of deep inelastic scattering (DIS) data and from a set of equal-time correlators computed using lattice QCD. We discuss our results, showing how the proposed methodology allows for a well-defined statistical interpretation of the different sources of errors entering the PDF uncertainty, and how results can be validated a posteriori.