We investigate the degree to which posterior expectations are sensitive to prior distributions, using a local method based on functional differentiation. Invariance considerations suggest a family of norms which can be used to measure perturbations to the prior. Our sensitivity measure is seen to depend heavily on the choice of norm; asymptotic results suggest which norm will yield the most useful results in practice. We find that to maintain asymptotically sensible behaviour, it is necessary to reduce the richness of the class of prior perturbations as the dimension of the parameter space increases. We give a characterization of Jeffreys' prior as the prior to which inference will be least sensitive.