  Computer experiments are becoming increasingly important in scientific
investigations. In the presence of uncertainty, analysts employ probabilistic
sensitivity methods to identify the key-drivers of change in the quantities of
interest. Simulation complexity, large dimensionality and long running times
may force analysts to make statistical inference at small sample sizes. Methods
designed to estimate probabilistic sensitivity measures at relatively low
computational costs are attracting increasing interest. We propose a fully
Bayesian approach to the estimation of probabilistic sensitivity measures based
on a one-sample design. We discuss, first, new estimators based on placing
piecewise constant priors on the conditional distributions of the output given
each input, by partitioning the input space. We then present two alternatives,
based on Bayesian non-parametric density estimation, which bypass the need for
predefined partitions. In all cases, the Bayesian paradigm guarantees the
quantification of uncertainty in the estimation process through the posterior
distribution over the sensitivity measures, without requiring additional
simulator evaluations. The performance of the proposed methods is compared to
that of traditional point estimators in a series of numerical experiments
comprising synthetic but challenging simulators, as well as a realistic
application. $\textit{An Updated Version of the Manuscript is Forthcoming in
Statistics and Computing.}$
