We present a Bayesian analysis of the combination of current neutrino oscillation, neutrinoless double beta decay and CMB observations. Our major goal is to carefully investigate the possibility to single out one neutrino mass ordering, Normal Ordering or Inverted Ordering, with current data. Two possible parametrizations (three neutrino masses versus the lightest neutrino mass plus the two oscillation mass splittings) and priors (linear versus logarithmic) are examined. We find that the preference for NO is only driven by neutrino oscillation data. Moreover, the values of the Bayes factor indicate that the evidence for NO is strong only when the scan is performed over the three neutrino masses with logarithmic priors; for every other combination of parameterization and prior, the preference for NO is only weak. As a by-product of our Bayesian analyses, we are able to a) compare the Bayesian bounds on the neutrino mixing parameters to those obtained by means of frequentist approaches, finding a very good agreement; b) determine that the lightest neutrino mass plus the two mass splittings parametrization, motivated by the physical observables, is strongly preferred over the three neutrino mass eigenstates scan and c) find that there is a weak-to-moderate preference for logarithmic priors. These results establish the optimal strategy to successfully explore the neutrino parameter space, based on the use of the oscillation mass splittings and a logarithmic prior on the lightest neutrino mass. We also show that the limits on the total neutrino mass $sum m_ u$ can change dramatically when moving from one prior to the other. These results have profound implications for future studies on the neutrino mass ordering, as they crucially state the need for self-consistent analyses which explore the best parametrization and priors, without combining results that involve different assumptions.
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