PDF reweighting in the Hessian matrix approach

Abstract

We introduce the Hessian reweighting of parton distribution functions (PDFs). Similarly to the better-known Bayesian methods, its purpose is to address the compatibility of new data and the quantitative modifications they induce within an existing set of PDFs. By construction, the method discussed here applies to the PDF fits that carried out a Hessian error analysis using a non-zero tolerance $\Delta\chi^2$. The principle is validated by considering a simple, transparent example. We are also able to establish an agreement with the Bayesian technique provided that the tolerance criterion is appropriately accounted for and that a purely exponential Bayesian likelihood is assumed. As a practical example, we discuss the inclusive jet production at the LHC.Comment: The final version, to appear in JHE

QCD analysis of W- and Z-boson production at Tevatron

“…The impact of the Tevatron data on the PDFs is studied with two different approaches: in the first approach they are included in a PDF fit together with the HERA I data, in the second, the impact on existing PDF sets is studied using Hessian profiling [19] and Bayesian reweighting techniques [20][21][22]. The PDF fit allows a detailed study of the settings of the QCD analysis, and in particular of various parametrisation forms.…”

“…The impact of a new data set on a given PDF set can be quantitatively estimated with a profiling procedure [19]. The profiling is performed using a χ 2 function which includes both the experimental uncertainties and the theoretical uncertainties arising from PDF variations:…”

“…In addition, the values at the minimum of the nuisance parameters β min k,th can be interpreted as optimisation ("profiling") of PDFs to describe the data [19]. Explicitly, the profiled central PDF set f 0 is given by…”

“…The Tevatron W -and Zboson measurements are also compared to predictions evaluated with the recent PDF sets CT10nlo [8], MMHT2014 [9] and NNPDF3.0 [10]. The impact of the Tevatron data on PDFs is studied using Hessian profiling [19] and Bayesian reweighting [20][21][22] techniques. The profiling of PDF uncertainties is generalised to the case of asymmetric PDF uncertainties.…”

Study of Jets Production Association with a Z boson in pp Collision at 7 and 8 TeV with the CMS Detector

“…One method of determining PDFs is based on a Hessian approach [79], which minimizes a suitable log-likelihood χ 2 function and accounts for correlated uncertainties by means of a covariance matrix. The best fit is the point in parameter space at which χ 2 is minimum.…”