Outsmarting Uncertainty

In this dissertation we explore three topics: 1) the extraction of the proton radius from electron scattering data; 2) the reconstruction of neutron distributions in nuclei from parity violating experiments; and 3) the propagation of dispersive corrections in electron-nucleus scattering to the estimated nuclear densities. We use various statistics tools and strategies including Bayesian analysis, bias-variance trade-off criteria, and a new framework we have developed under the name of ``the transfer function formalism". The first topic is motivated by the decade old ``proton radius puzzle": a stark disagreement between muonic based and electronic based experiments aimed to extract the electric radius of the proton. Given the long and impassioned discussion about what model should be used to recover the radius, the main question driving us is: What is the most robust and reliable way, from a statistical point of view, to extrapolate the electron scattering data and extract the radius? The second topic is motivated by the combination of the strong interest in weak charge densities by the nuclear physics and astrophysics communities, and the enormously challenging experiments needed to access neutron distributions inside a nucleus. Given the complexity and cost of such experiments, the main question driving us is: How to extract the most amount of information in cases with scarce data? The third topic is motivated by our lack of knowledge on the magnitude and energy dependence of dispersive effects, a fact that could negatively impact delicate experiments such as those involved with the second topic. In addition, the propagation of dispersive effects from raw data, to the calculated quantities through the fitted model, gave us a perfect benchmark to implement our proposed transfer function formalism. Given the uncertain structure of the dispersive effects, the main question driving us is: How to propagate small perturbations in the data to calculated quantities in a straightforward way (without re-fitting the model)? This dissertation is a contribution to address these questions, and to strengthen multidisciplinary efforts between physics and statistics.