Following the critical review of Seaman et al. (2012), we reflect on what is presumably the most essential aspect of Bayesian statistics, namely the selection of a prior density. In some cases, Bayesian inference remains fairly stable under a large range of noninformative prior distributions. However, as discussed by \citet{Hd}, there may also be unintended consequences of a choice of a noninformative prior and, these authors consider this problem ignored in Bayesian studies. As they based their argumentation on four examples, we reassess these examples and their Bayesian processing via different prior choices. Our conclusion is to lower the degree of worry about the impact of the prior, exhibiting an overall stability of the posterior distributions. We thus consider that the warnings of Seaman et al. (2012), while commendable, do not jeopardize the use of most noninformative priors.