  Robust Bayesian linear regression is a classical but essential statistical
tool. Although novel robustness properties of posterior distributions have been
proved recently under a certain class of error distributions, their sufficient
conditions are restrictive and exclude several important situations. In this
work, we revisit a classical two-component mixture model for response
variables, also known as contamination model, where one component is a
light-tailed regression model and the other component is heavy-tailed. The
latter component is independent of the regression parameters, which is crucial
in proving the posterior robustness. We obtain new sufficient conditions for
posterior (non-)robustness and reveal non-trivial robustness results by using
those conditions. In particular, we find that even the Student-$t$ error
distribution can achieve the posterior robustness in our framework. A numerical
study is performed to check the Kullback-Leibler divergence between the
posterior distribution based on full data and that based on data obtained by
removing outliers.
