  We discuss how to apply the Hessian method (i) to predict the impact of a new
data set (or sets) on the parton distribution functions (PDFs) and their
errors, by producing an updated best-fit PDF and error PDF sets, such as the
CTEQ-TEA PDFs; (ii) to predict directly the effect of a new data set on the PDF
errors of any other set of observables, without the need to recalculate using
the new error PDFs; and (iii) to transform the original set into a reduced set
of error PDFs which is optimized for a specific set of observables to reproduce
the PDF-induced uncertainties to any specified precision. We present a software
package, {\tt ePump} (error PDF Updating Method Package), that can be used to
update or optimize a set of PDFs, including the best-fit PDF set and Hessian
eigenvector pairs of PDF sets (i.e., error PDFs), and to update any other set
of observables. We demonstrate the potential of the program by presenting
selected phenomenological applications relevant to the Large Hadron Collider.
Special care is given to discuss the assumptions made and the limitations of
this theoretical framework compared to a treatment by the full global analysis
program.
