  The neutron distribution of neutron-rich nuclei provides critical information
on the structure of finite nuclei and neutron stars. Parity violating
experiments -- such as PREX and CREX -- provide a clean and largely
model-independent determination of neutron densities. Such experiments,
however, are challenging and expensive which is why sound statistical arguments
are required to maximize the information gained. For this goal we introduce a
new framework, "the transfer function formalism", aimed at uncertainty
quantification, model selection, and experimental design in the context of
neutron densities. The transfer functions (TFs) are built analytically by
expressing the linear response of the objective function to small perturbations
of the data. Using the TF formalism, we are able to analyze the expected
overall uncertainty -- quantified in terms of bias and variance -- of the mean
square radius and interior density of $^{48}$Ca and $^{208}$Pb. Using
relativistic mean field models as a proxy for the weak-charge density -- and
assuming that a total of five measurements could be performed on the weak form
factor of $^{48}$Ca and $^{208}$Pb -- we identify the optimal models and
experimental locations that minimize the combined radius and interior
uncertainty for both nuclei. We also explore the use of the TF formalism to
understand the influence of prior distributions for the model parameters, as
well as the optimization of model hyperparameters not constrained by the data.
