  We show how to account for correlations between theoretical uncertainties
incorporated in parton distribution function (PDF) fits, and the theoretical
uncertainties in the predictions made using these PDFs. We demonstrate by
explicit calculations, both analytical and numerical, that these correlations
can lead to corrections to the central values of the predictions, and
reductions in both the PDF uncertainties and the theoretical uncertainties in
the prediction. We illustrate our results with predictions for top production
rapidity distributions and the Higgs total cross-section at the LHC, using the
NLO NNPDF3.1 PDF set which incorporates missing higher order uncertainties. We
conclude that the inclusion of correlations can increase both the accuracy and
precision of predictions involving PDFs, particularly for processes with data
already included in the PDF fit.
