Bayesian approach to inverse problems. An application to NNPDF closure testing

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), we discuss some analytical results that can be obtained for linear problems and use these results as a guidance for 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. 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 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.