Information Criteria for Selecting Parton Distribution Function Solutions

TL;DR

This paper introduces information-theoretic and optimal-transport algorithms to classify, cluster, and select solutions for Parton Distribution Functions, addressing the challenge of multiple acceptable solutions in global QCD analyses.

Contribution

It proposes novel algorithms using Rényi entropy and Wasserstein distance for solution classification and selection in PDF determination, emphasizing shape characterization.

Findings

Rényi entropy effectively characterizes solution space.

Pareto fronts help identify optimal solution combinations.

Rényi entropy is versatile for clustering applications.

Abstract

In data-driven determination of Parton Distribution Functions (PDFs) in global QCD analyses, uncovering the true underlying distributions is complicated by a highly convoluted inverse problem. The determination of PDFs can be understood as the inference of a function supported on $[0,1]$, a problem that admits multiple acceptable solutions. An ensemble of solutions exists that pass all standard goodness-of-fit criteria. In this paper, we propose algorithms for the classification, clustering, and selection of solutions to the determination of PDFs, or any functions on $[0,1]$, based on the characterization of their shape. We explore information-theoretic based (R\'enyi entropy and divergence) and optimal-transport based (Wasserstein distance) criteria. In particular, we advocate for the use of the R\'enyi entropy as an {\it absolute} estimator per solution, as opposed to {\it relative} estimators that compare solutions pairwise. We show that the R\'enyi entropy can characterize the space of solutions {\it w.r.t.} the PDF shapes. Paired with the identification of the optimal combination of solutions via Pareto fronts, it provides a plausible and minimalist selection algorithm. Moreover, R\'enyi entropy proves versatile for use in clustering applications.