Hessian PDF reweighting meets the Bayesian methods
We discuss the Hessian PDF reweighting — a technique intended to estimate the effects that new measurements have on a set of PDFs. The method stems straightforwardly from considering new data in a usual -fit and it naturally incorporates also non-zero values for the tolerance, . In comparison to the contemporary Bayesian reweighting techniques, there is no need to generate large ensembles of PDF Monte-Carlo replicas, and the observables need to be evaluated only with the central and the error sets of the original PDFs. In spite of the apparently rather different methodologies, we find that the Hessian and the Bayesian techniques are actually equivalent if the criterion is properly included to the Bayesian likelihood function that is a simple exponential.
Hessian PDF reweighting meets the Bayesian methods
\FullConferenceXXII. International Workshop on Deep-Inelastic Scattering and Related Subjects,
28 April - 2 May 2014
The flood of hard-process data from the LHC proton+proton collisions that can test and offer further constraints for the parton distribution functions (PDFs) is nowadays so massive that the need to efficiently quantify the implications of different measurements has called for novel analysis techniques. To this end, an option that has gained some popularity is to make use of HERAFitter  to check the constraining power of the new data. However, in most cases this has meant comparing the PDFs obtained by using only the HERA deep-inelastic scattering data with the ones including additionally a specific set of LHC data (see e.g. [2, 3]). Clearly, there is no guarantee that this would reflect the impact of the new data in the global context. In this case the PDF reweighting methods [4, 5, 6, 7, 8], discussed in this talk, should be more adequate.
2 The Hessian reweighting
Let us suppose we have a set of Hessian PDFs with a global tolerance . The PDFs have been parametrized by some fixed functional form and the central set corresponds to those parameter values that minimize a global -function. The Hessian procedure  to quantify the PDF errors is based on expanding this -function around the minimum with respect to the fit parameters and diagonalizing the Hessian matrix :
The coordinates of the central set and error sets in this -space (“space of eigenvectors”) are
and estimate the PDF-dependent theory values by a linear approximation as
The function is thus a second-order polynomial in variables and its minimum occurs at with
The corresponding new PDFs (omitting here all arguments and flavor indices) are easily obtained, by the same approximation as in Eq. (4),
and by rewriting the function as one can also define the new PDF error sets by the same procedure as above. The increase of the original — the “reweighting penalty” — can be approximated by
3 The Bayesian procedures
The Bayesian PDF reweighting methods data back to the original works of Giele and Keller  and were later on revived by the NNPDF collaboration [4, 5]. To apply these techniques in the case of Hessian PDFs one constructs an ensemble of PDF replicas by 
where are Gaussian random numbers. The PDF dependent observables are then obtained as expectation values
which coincide with if the non-linearities are small and the number of replicas is sufficiently large. The Bayesian reweighting amounts to turning these averages to weighted ones
where the weights are determined solely from the new data. Two different functional forms have appeared in the literature: The one proposed originally by Giele and Keller is a simple exponential
The reweighting penalty can be computed by
4 Simplified example
To compare the different reweighting methods we invoke a simple example by constructing two sets of pseudodata for a function shown in Figure 1. We use the first one to construct a set of “Hessian PDFs” as outlined in Section 2 (using the same functional form for ) with a chosen tolerance . Then we take the second data set, work out the predictions of the reweighting methods and compare those to a direct fit including both of these data sets.
The results of this exercise are shown in Figure 2. We observe that in the case the Hessian and Giele-Keller reweighting are in perfect agreement with the direct fit (left-hand panel). If we increase the tolerance to , the Hessian procedure still accords with the direct fit but the Giele-Keller method appears to fail (middle panel). However, the agreement can be easily restored by rescaling the values of in Eq. (14) as (left-hand panel). In all cases the Bayesian weights which have been shown to work for the NNPDF-style fits (the chi-squared weights) yield clearly different results.
5 CTEQ6.6 and inclusive jets at the LHC
Having now understood how to correctly reweight Hessian PDFs, we illustrate what would be the effect of LHC inclusive jet data on the CTEQ6.6 PDFs  (for which which ). Specifically, we focus on the jet measurements by the CMS collaboration  and use the FASTNLO interface [13, 14, 15] for the computations. Before the reweighting CTEQ6.6 tends to somewhat overpredict the experimental cross sections as shown in Figure 3 (left-hand panel) which, however, largely disappears after applying the correlated systematic shifts (right-hand panel). Initially, (for data points).
The gluon distributions after applying the reweighting procedures are presented in Figure 4 revealing a decrease in the large- gluon PDF. As expected, the Hessian and (rescaled) Giele-Keller reweighting agree and only a modest penalty of units is induced. The new global has changed by units. The result of Bayesian reweighting with chi-squared weights is shown for comparison and a similar but much too pronounced effect is observed. In fact, instead of decreasing, the new global has increased by units. This is also reflected in the new cross-section predictions shown in Figure 5: While the Hessian reweighting (left-hand panel) predicts only a modest decrease in the cross sections (moderating the overshooting observed in Figure 5), the Bayesian reweighting with chi-squared weights (right-hand panel) would lead us to believe in much larger effect.
We have discussed how to estimate the effects that a new set of data would have on a global Hessian PDF fit with fixed tolerance . By considering a simple example, we find that there are two alternative techniques that give essentially the same answer and are equivalent to a direct refit: the Hessian reweighting and a Bayesian technique with rescaled Giele-Keller weights. As a practical example, we employed these methods in the case of inclusive jet production at the LHC.
H.P. wants to acknowledge the financial support from the Academy of Finland, Project No. 133005. P.Z. is supported by European Research Council grant HotLHC ERC-2011-StG-279579; by Ministerio de Ciencia e Innovación of Spain under project FPA2011-22776, and the Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042); by Xunta de Galicia (GRC2013-024); and by FEDER.
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