Gaussian Processes for Inferring Parton Distributions

The extraction of parton distribution functions (PDFs) from experimental or lattice QCD data is an ill-posed inverse problem, where regularization strongly impacts both systematic uncertainties and the reliability of the results. We study a framework based on Gaussian Process Regression (GPR) to reconstruct PDFs from lattice QCD matrix elements. Within a Bayesian framework, Gaussian processes serve as flexible priors that encode uncertainties, correlations, and constraints without imposing rigid functional forms. We investigate a wide range of kernel choices, mean functions, and hyperparameter treatments. We quantify information gained from the data using the Kullback Leibler divergence. Synthetic data tests demonstrate the consistency and robustness of the method. Our study establishes GPR as a systematic and non-parametric approach to PDF reconstruction, offering controlled uncertainty estimates and reduced model bias in lattice QCD analyses.

💡 Research Summary

The paper addresses the long‑standing inverse problem of extracting parton distribution functions (PDFs) from a finite set of noisy data, focusing on matrix elements computed in lattice QCD (LQCD). Traditional approaches either impose a low‑dimensional parametric form (e.g., N x^α(1−x)^β) or use neural‑network parameterizations, both of which introduce model bias and require ad‑hoc regularization. The authors propose a fully Bayesian, non‑parametric framework based on Gaussian Process Regression (GPR). In this setting the prior over the unknown function q(x) is a Gaussian process defined by a mean function μ(x) and a covariance kernel k(x,x′). By choosing appropriate kernels and means, one can encode physical constraints such as positivity, normalization, and the expected vanishing of the PDF at x = 1, while retaining maximal flexibility.

Key methodological steps include:

1. Formal Bayesian formulation – The data M(ν) are related to q(x) through an integral kernel B(ν,x) (essentially a Fourier cosine/sine transform for pseudo‑PDFs). The likelihood is taken Gaussian (χ²), and the posterior P