Fitting parton distribution data with multiplicative normalization uncertainties

The extraction of robust parton distribution functions with faithful errors requires a careful treatment of the uncertainties in the experimental results. In particular, the data sets used in current analyses each have a different overall multiplicative normalization uncertainty that needs to be properly accounted for in the fitting procedure. Here we consider the generic problem of performing a global fit to many independent data sets each with a different overall multiplicative normalization uncertainty. We show that the methods in common use to treat multiplicative uncertainties lead to systematic biases. We develop a method which is unbiased, based on a self-consistent iterative procedure. We then apply our generic method to the determination of parton distribution functions with the NNPDF methodology, which uses a Monte Carlo method for uncertainty estimation.