In proton-nucleus and nucleus-nucleus collision experiments, one determines the centrality of a collision according to the multiplicity or energy deposited in a detector. This serves as a proxy for the true collision centrality, as defined by the impact parameter. We show that the probability distribution of impact parameter in a given bin of experiment-defined centrality can be reconstructed without assuming any specific model for the collision dynamics, in both proton-nucleus and nucleus-nucleus systems. The reconstruction is reliable up to about 10\% centrality, and is more accurate for nucleus-nucleus collisions. We perform an application of our procedure to experimental data from all the CERN Large Hadron Collider (LHC) collaborations, from which we extract, in Pb+Pb and $p$+Pb collisions, the corresponding distributions of impact parameter.

11 pages, 6 figures; v2: published version. Improved explanation of the equivalence between gamma and negative binomial distributions in footnote 1. Ancillary files contain the fits of trento data, and the reconstructions of impact parameter distributions, using the gamma fluctuation kernel