  We propose a general solution to the problem of robust Bayesian inference in
complex settings where outliers may be present. In practice, the automation of
robust Bayesian analyses is important in the many applications involving large
and complex datasets. The proposed solution relies on a reformulation of
Bayesian inference based on possibility theory, and leverages the observation
that, in this context, the marginal likelihood of the data assesses the
consistency between prior and likelihood rather than model fitness. Our
approach does not require additional parameters in its simplest form and has a
limited impact on the computational complexity when compared to non-robust
solutions. The generality of our solution is demonstrated via applications on
simulated and real data including matrix estimation and change-point detection.
