We study the Full Bayesian Updating rule for convex capacities. Following a route suggested by Jaffray (IEEE Transactions on Systems, Man and Cybernetics 22(5):1144–1152, 1992), we define some properties one may want to impose on the updating process, and identify the classes of (convex and strictly positive) capacities that satisfy these properties for the Full Bayesian Updating rule. This allows us to characterize two parametric families of convex capacities: \({(\varepsilon,\delta)}\) -contaminations (which were introduced, in a slightly different form, by Huber (Robust Statistics, Wiley, New York, 1981)) and \({\varepsilon}\) -contaminations.