Gaussian Process à la Feynman: Extracting PDF from LQCD

We present a Bayesian reconstruction of parton distribution functions (PDFs) from Lattice QCD data using Gaussian Processes (GPs). Within the pseudo-distribution framework, matrix elements of bilocal operators can be computed off the light cone and related to PDFs via a Fourier transform to the so called Ioffe time distribution, which directly represent the Lattice QCD meassurements performed in Euclidean space. This setup transforms the problem of determining PDFs into an inverse one: we aim to recover a smooth, continuous distribution from a finite set of noisy and correlated data points. This inverse problem is inherently ill-posed, and its solution requires careful regularization.

Gaussian Processes provide a robust and flexible tool to quantify uncertainty, offering a non-parametric prior over functions and enabling the incorporation of prior physical knowledge such as positivity, endpoint behavior, and sum rules. We explore different kernel-mean structures and hyperparameter strategies to control the bias–variance trade-off. Additionally, we implement the Kullback–Leibler divergence to quantify information gain by comparing posterior and prior distributions globally and point by point. Finally, we integrate the models explored into a model selection and averaging procedure to obtain a more robust and interpretable reconstruction of PDFs, grounded in modern probabilistic inference.