The Local Sensitivity of Posterior Expectations
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.
Key Words: Classes of probabilities, Fréchet derivative,
Jeffreys' prior, local sensitivity, -norm, robustness.