  Robust Bayesian analysis has been mainly devoted to detecting and measuring
robustness to the prior distribution. Indeed, many contributions in the
literature aim to define suitable classes of priors which allow the computation
of variations of quantities of interest while the prior changes within those
classes. The literature has devoted much less attention to the robustness of
Bayesian methods to the likelihood function due to mathematical and
computational complexity, and because it is often arguably considered a more
objective choice compared to the prior. In this contribution, a new approach to
Bayesian local robustness to the likelihood function is proposed and extended
to robustness to the prior and to both. This approach is based on the notion of
distortion function introduced in the literature on risk theory, and then
successfully adopted to build suitable classes of priors for Bayesian global
robustness to the prior. The novel robustness measure is a local sensitivity
measure that turns out to be very tractable and easy to compute for certain
classes of distortion functions. Asymptotic properties are derived and
numerical experiments illustrate the theory and its applicability for modelling
purposes.
