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Determination of proton PDF uncertainties with Markov chain Monte Carlo

P. Risse1,2,3,*, N. Derakhshanian4, T. Ježo1,†, K. Kovařík1, and A. Kusina4,‡

  • *Contact author: prisse@smu.edu
  • †Contact author: tomas.jezo@uni-muenster.de
  • ‡Contact author: aleksander.kusina@ifj.edu.pl
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Phys. Rev. D 113, 054049 – Published 27 March, 2026

DOI: https://doi.org/10.1103/5y5y-5yhy

Abstract

We present an analysis of parton distribution functions (PDFs) of the proton using Markov chain Monte Carlo (MCMC) methods. The MCMC approach naturally implements Bayes’ theorem and, thus, provides a means to directly sample the underlying probability distribution—in this case, the probability distribution of the PDF parameters. This allows for a straightforward propagation of the resulting uncertainties into any PDF-dependent observable, preserving their simple probabilistic interpretation. In our analysis we include a broad set of deep inelastic scattering data from HERA, BCDMS and NMC experiments along with the Drell-Yan, W and Z boson data from LHC and Tevatron experiments, which combined with theoretical calculations at next-to-next-to-leading order in QCD allow for realistic determination of PDFs. The main focus of this analysis is to explore alternative methods for PDF uncertainty estimation that are more firmly grounded in statistical principles. We show that the flexibility of the Bayes framework, allowing one, e.g., to account for non-Gaussianity or inconsistencies of datasets, is crucial to extract realistic uncertainties when such assumptions are not fulfilled. We also demonstrate that MCMC allows one to determine the Δχ2 value corresponding to a given confidence level in the sample, which can, in turn, be used as a statistically well-founded tolerance criterion used in the Hessian method, thus addressing one of its main long-standing drawbacks.

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Phys. Rev. D 113, 054049– Published 27 March, 2026
  • Received 4 November 2025
  • Accepted 25 February 2026
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