Information criteria for selecting parton distribution function solutions

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 (Rényi entropy and divergence) and optimal-transport based (Wasserstein distance) criteria. In particular, we advocate for the use of the Rényi entropy as an absolute estimator per solution, as opposed to relative estimators that compare solutions pairwise. We show that the Rényi entropy can characterize the space of solutions 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, Rényi entropy proves versatile for use in clustering applications.