Radial distribution functions (RDFs) are fundamental to evaluating molecular models of liquids. Yet a critical aspect often overlooked in the literature is the accurate quantification of RDF uncertainty arising from noisy and uncertain experimental structure factor measurements. In general, extracting RDFs from scattering data has two restrictive bottlenecks: (1) the decomposition of the total structure factor into site-site partial structure factors is ill-posed, and (2) the necessary Fourier transform from momentum- into real-space is sensitive to binning and windowing artifacts. To address these issues, we develop a physics-informed, non-stationary Gaussian process framework that rigorously quantifies and propagates uncertainty from experimental scattering data to site-site partial RDFs. The approach is built around a non-stationary Gibbs kernel, which incorporates known physical constraints on RDF behavior such as vanishing at short distances, convergence to unity at long range, and the presence of molecular bonds. We then apply this method to neutron and X-ray scattering data of liquid water and show that it produces uncertainty estimates that are more physically consistent than those obtained from current state-of-the-art techniques. The result is a new benchmark for liquid water structure, and more broadly, a principled framework for incorporating experimental uncertainty into molecular model evaluation.
