Bayesian Approach to Inverse Problems: an Application to NNPDF Closure Testing

Abstract

We discuss the Bayesian approach to the solution of inverse problems and apply the formalism to analyse the closure tests performed by the NNPDF collaboration. Starting from a comparison with the approach that is currently used for the determination of parton distributions (PDFs) by the NNPDF collaboration, we discuss some analytical results that can be obtained for linear problems and use these results as a guidance for the more complicated non-linear problems. We show that, in the case of Gaussian distributions, the posterior probability density of the parametrized PDFs is fully determined by the results of the NNPDF fitting procedure. In the particular case that we consider, the fitting procedure and the Bayesian analysis yield exactly the same result. Building on the insight that we obtain from the analytical results, we introduce new estimators to assess the statistical faithfulness of the fit results in closure tests. These estimators are defined in data space, and can be studied analytically using the Bayesian formalism in a linear model in order to clarify their meaning. Finally we present numerical results from a number of closure tests performed with current NNPDF methodologies. These further tests allow us to validate the NNPDF4.0 methodology and provide a quantitative comparison of the NNPDF4.0 and NNPDF3.1 methodologies. As PDFs determinations move into precision territory, the need for a careful validation of the methodology becomes increasingly important: the error bar has become the focal point of contemporary PDFs determinations. In this perspective, theoretical assumptions and other sources of error are best formulated and analysed in the Bayesian framework, which provides an ideal language to address the precision and the accuracy of current fits.

References in corpus (9)

• The Asymmetry of Antimatter in the Proton
• nNNPDF2.0: Quark Flavor Separation in Nuclei from LHC Data
• Measurement of the inclusive jet cross-sections in proton--proton collisions at $\sqrt{s}=$8 TeV with the ATLAS detector
• The Strangest Proton?
• Measurement of top quark pair differential cross-sections in the dilepton channel in $pp$ collisions at $\sqrt{s}$ = 7 and 8 TeV with ATLAS
• Measurement of the forward Z boson production cross-section in pp collisions at $\sqrt{s}=13$ TeV
• Measurement of the $t\bar{t}$ production cross-section in the lepton+jets channel at $\sqrt{s}=13$TeV with the ATLAS experiment
• Deuteron Uncertainties in the Determination of Proton PDFs
• Future tests of parton distributions

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