  The topic of robustness is experiencing a resurgence of interest in the
statistical and machine learning communities. In particular, robust algorithms
making use of the so-called median of means estimator were shown to satisfy
strong performance guarantees for many problems, including estimation of the
mean, covariance structure as well as linear regression. In this work, we
propose an extension of the median of means principle to the Bayesian
framework, leading to the notion of the robust posterior distribution. In
particular, we (a) quantify robustness of this posterior to outliers, (b) show
that it satisfies a version of the Bernstein-von Mises theorem that connects
Bayesian credible sets to the traditional confidence intervals, and (c)
demonstrate that our approach performs well in applications.
