The choice of data that enters a global QCD analysis can have a substantial impact on the resulting parton distributions and their predictions for collider observables. One of the main reasons for this has to do with the possible presence of inconsistencies, either internal within an experiment or external between different experiments. In order to assess the robustness of the global fit, different definitions of a conservative PDF set, that is, a PDF set based on a maximally consistent dataset, have been introduced. However, these approaches are typically affected by theory biases in the selection of the dataset. In this contribution, after a brief overview of recent NNPDF developments, we propose a new, fully objective, definition of a conservative PDF set, based on the Bayesian reweighting approach. Using the new NNPDF3.0 framework, we produce various conservative sets, which turn out to be mutually in agreement within the respective PDF uncertainties, as well as with the global fit. We explore some of their implications for LHC phenomenology, finding also good consistency with the global fit result. These results provide a non-trivial validation test of the new NNPDF3.0 fitting methodology, and indicate that possible inconsistencies in the fitted dataset do not affect substantially the global fit PDFs.