This paper deals with measures of local robustness for particular Bayesian quantities, i.e. posterior summaries. We build a framework where any Bayesian quantity can be seen as a posterior functional and its sensitivity to all inputs is checked. First, we use the Gateaux derivatives to measure the impact on posterior summaries of perturbations of prior or sampling models, giving some general expressions. Such quantities capture both a ’data effect’ and a ’model effect’ on the functional. Secondly, we check the sensitivity to one observation in the sample, once a particular combination of prior/sampling models has been chosen. Moreover, we propose a new estimator of the Bayes factor for practical implementation. Finally, illustrative examples on sensitivity analysis are provided and discussed.
