  Bayesian inference and Gaussian processes are widely used in applications
ranging from robotics and control to biological systems. Many of these
applications are safety-critical and require a characterization of the
uncertainty associated with the learning model and formal guarantees on its
predictions. In this paper we define a robustness measure for Bayesian
inference against input perturbations, given by the probability that, for a
test point and a compact set in the input space containing the test point, the
prediction of the learning model will remain $\delta-$close for all the points
in the set, for $\delta>0.$ Such measures can be used to provide formal
guarantees for the absence of adversarial examples. By employing the theory of
Gaussian processes, we derive tight upper bounds on the resulting robustness by
utilising the Borell-TIS inequality, and propose algorithms for their
computation. We evaluate our techniques on two examples, a GP regression
problem and a fully-connected deep neural network, where we rely on weak
convergence to GPs to study adversarial examples on the MNIST dataset.
