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.

A new generation of simultaneous fits to LHC data using deep learning

“…The PDF side of SIMUnet is of course identical to NNPDF4.0 which has been demonstrated to be able to replicate an underlying law using the closure testing framework [2,72]. In this section we will fix the value of the Wilson coefficients a priori and study how effectively we can retrieve these values using our methodology.…”

“…In this section, we assess the robustness of our approach through the closure testing framework defined in Refs. [71,72]. Here we do not address the robustness of the PDF part of the fit, which in a sense comes from the results presented in the NNPDF4.0 paper [2] and the following dedicated study of Ref.…”

“…Here we do not address the robustness of the PDF part of the fit, which in a sense comes from the results presented in the NNPDF4.0 paper [2] and the following dedicated study of Ref. [72], but rather focus on the robustness of the fit of the Wilson coefficients and of the PDFs in a simultaneous fit.…”

Determination of the theory uncertainties from missing higher orders on NNLO parton distributions with percent accuracy

“…the recent studies in Refs. [2,3]). However, PDF uncertainties, as given in all standard PDF sets, such as NNPDF4.0 [1], CT18 [4], MSHT20 [5] or ABMP16 [6], do not include theoretical uncertainties, i.e., the uncertainties that affect the predictions that are compared to the data in the process of determining PDFs from a set of experimental data.…”

Bayesian inference for form-factor fits regulated by unitarity and analyticity

“…We note that a number of novel ideas applying Bayesian inference in the context of quantum-field theory have recently been put forward in a variety of contexts: fitting of parton-distribution functions [15], analysis of fits to lattice data [16][17][18], or the estimation of missing higher-order terms in perturbation theory [19].…”