  We propose here a \emph{novel} method which singles out the \emph{a priori}
unavoidable dependence on the underlying cosmological model when extracting
parameter constraints, providing robust limits which only depend on the
considered dataset. Interestingly, when dealing with several possible
cosmologies and interpreting the Bayesian preference in terms of the Gaussian
statistical evidence, the preferred model is much less favored than when only
two cases are compared. As a working example, we apply our approach to the
cosmological neutrino mass bounds, which play a fundamental role not only in
establishing the contribution of relic neutrinos to the dark matter of the
Universe, but also in the planning of future experimental searches of the
neutrino character and of the neutrino mass ordering.
