Hessian PDF reweighting meets the Bayesian methods

TL;DR

This paper compares Hessian PDF reweighting with Bayesian methods, showing they are equivalent when the $hi^2$ criterion is properly included, offering a computationally efficient approach to updating PDFs with new data.

Contribution

It demonstrates the equivalence of Hessian and Bayesian reweighting techniques when incorporating the $hi^2$ criterion, simplifying PDF updates without large Monte Carlo ensembles.

Findings

Hessian and Bayesian methods are equivalent with proper $hi^2$ inclusion.

Hessian reweighting is computationally more efficient.

The approach naturally includes non-zero tolerance values.

Abstract

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 $\chi^2$-fit and it naturally incorporates also non-zero values for the tolerance, $\Delta\chi^2>1$. 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 $\Delta\chi^2$ criterion is properly included to the Bayesian likelihood function that is a simple exponential.