Bayesian inference with Gaussian processes for the determination of parton distribution functions
Alessandro Candido (Theoretical Physics Department, CERN, Geneva 23, 1211, Switzerland); Luigi Debbio (Higgs Centre for Theoretical Physics, School of Physics and Astronomy, Peter Guthrie Tait Road, Edinburgh, EH9 3 FD, UK); Tommaso Giani (Nikhef Theory Group, Science Park 105, Amsterdam, 1098 XG, The Netherlands, Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, 1081 HV, The Netherlands); Giacomo Petrillo (Dipartimento di Statistica, Informatica, Applicazioni “Giuseppe Parenti” (DISIA), Università di Firenze, Viale Morgagni 59, Firenze, 50134, Italy)
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.
