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  • Open Access

Determination of uncertainties in parton densities

N. T. Hunt-Smith, A. Accardi, W. Melnitchouk, N. Sato, A. W. Thomas, and M. J. White
Phys. Rev. D 106, 036003 – Published 2 August 2022
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  • INTRODUCTION
  • FORMALISM
  • COMPARING METHODOLOGIES
  • NEURAL NETWORKS, PARTITIONING, AND CROSS…
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We review various methods used to estimate uncertainties in quantum correlation functions, such as parton distribution functions (PDFs). Using a toy model of a PDF, we compare the uncertainty estimates yielded by the traditional Hessian and data resampling methods, as well as from explicitly Bayesian analyses using nested sampling or hybrid Markov chain Monte Carlo techniques. We investigate how uncertainty bands derived from neural network approaches depend on details of the network training, and how they compare to the uncertainties obtained from more traditional methods with a specific underlying parametrization. Our results show that utilizing a neural network on a simplified example of PDF data has the potential to inflate uncertainties, in part due to the cross-validation procedure that is generally used to avoid overfitting data.

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    • Received 11 July 2022
    • Accepted 14 July 2022

    DOI:https://doi.org/10.1103/PhysRevD.106.036003

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Parton distribution functions
    Particles & Fields

    Authors & Affiliations

    N. T. Hunt-Smith1, A. Accardi2,3, W. Melnitchouk3, N. Sato3, A. W. Thomas1, and M. J. White1

    • 1CSSM and ARC Centre of Excellence for Dark Matter Particle Physics, Department of Physics, The University of Adelaide, Adelaide 5005, Australia
    • 2Hampton University, Hampton, Virginia 23668, USA
    • 3Jefferson Lab, Newport News, Virginia 23606, USA

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    Issue

    Vol. 106, Iss. 3 — 1 August 2022

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    Images

    • Figure 1
      Figure 1

      Left: quark distributions q1 and q2 in our toy model as a function of x. Right: example of toy cross-section datasets σ1 and σ2 for the pseudodata (filled and open circles) and cross sections calculated from the input q1 and q2 quark distributions (blue and orange curves).

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    • Figure 2
      Figure 2

      Comparison of fit results for the DR, HMC, NS, and Hessian methods. The top row contains cross-section and PDF distributions, while the middle and bottom rows show ratios to the true values for each distribution.

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    • Figure 3
      Figure 3

      An example of an NN overfitting a single set of our toy data.

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    • Figure 4
      Figure 4

      Comparison of training (blue) and validation (green) set loss functions for an overfitting example.

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    • Figure 5
      Figure 5

      Comparison of the σ1 replica set for the NN fit with (red band) and without (yellow band) cross validation.

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    • Figure 6
      Figure 6

      Variation of the 1σ uncertainty with x across several f values from 0.1 to 0.9 for a single dataset of 50 points.

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    • Figure 7
      Figure 7

      Total uncertainty ratio to f=0.5 for different partition fractions as a function of number of data points.

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    • Figure 8
      Figure 8

      Comparison of a single σ1 replica set between NN and parametric DR methods (left panel), and the ratio of each predicted point to the true underlying law (right panel).

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    • Figure 9
      Figure 9

      Study of confidence interval coverage for the NN (red lines) and parametric DR (blue lines) methods as a function of x.

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