  Determinations of the proton's collinear parton distribution functions (PDFs)
are emerging with growing precision due to increased experimental activity at
facilities like the Large Hadron Collider. While this copious information is
valuable, the speed at which it is released makes it difficult to quickly
assess its impact on the PDFs, short of performing computationally expensive
global fits. As an alternative, we explore new methods for quantifying the
potential impact of experimental data on the extraction of proton PDFs. Our
approach relies crucially on the Hessian correlation between theory-data
residuals and the PDFs themselves, as well as on a newly defined quantity ---
the sensitivity --- which represents an extension of the correlation and
reflects both PDF-driven and experimental uncertainties. This approach is
realized in a new, publicly available analysis package PDFSense, which operates
with these statistical measures to identify particularly sensitive experiments,
weigh their relative or potential impact on PDFs, and visualize their detailed
distributions in a space of the parton momentum fraction $x$ and factorization
scale $\mu$. This tool offers a new means of understanding the influence of
individual measurements in existing fits, as well as a predictive device for
directing future fits toward the highest impact data and assumptions. Along the
way, many new physics insights can be gained or reinforced. As one of many
examples, PDFSense is employed to rank the projected impact of new LHC
measurements in jet, vector boson, and $t\bar{t}$ production and leads us to
the conclusion that inclusive jet production at the LHC has a potential for
playing an indispensable role in future PDF fits. These conclusions are
independently verified by preliminarily fitting this experimental information
and investigating the constraints they supply using the Lagrange multiplier
technique.
