  In this paper we revisit the question of the information which cosmology
provides on the scenarios with sterile neutrinos invoked to describe the SBL
anomalies using Bayesian statistical tests. We perform an analysis of the
cosmological data in $\Lambda$CDM$+r+\nu_s$ cosmologies for different
cosmological data combinations, and obtain the marginalized cosmological
likelihood in terms of the two relevant parameters, the sterile neutrino mass
$m_s$ and its contribution to the energy density of the early Universe $N_{\rm
eff}$. We then present an analysis to quantify at which level a model with one
sterile neutrino is (dis)favoured with respect to a model with only three
active neutrinos, using results from both short-baseline experiments and
cosmology. We study the dependence of the results on the cosmological data
considered, in particular on the inclusion of the recent BICEP2 results and the
SZ cluster data from the Planck mission. We find that only when the cluster
data is included the model with one extra sterile neutrino can become more
favoured that the model with only the three active ones provided the sterile
neutrino contribution to radiation density is suppressed with respect to the
fully thermalized scenario. We have also quantified the level of
(in)compatibility between the sterile neutrino masses implied by the
cosmological and SBL results.
