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.

翻译：稳健贝叶斯线性回归是一项经典但基础的统计工具。尽管近年来在特定误差分布类别下，后验分布的新颖稳健性性质已被证明，但其充分条件具有限制性，并排除了若干重要情形。本文重新审视了响应变量的一种经典双组分混合模型（亦称污染模型），其中一组分为轻尾回归模型，另一组分为重尾模型。后者独立于回归参数，这一性质对证明后验稳健性至关重要。我们获得了后验（非）稳健性的新充分条件，并利用这些条件揭示了非平凡的稳健性结果。特别地，本研究发现，即使学生-t误差分布也能在我们的框架中实现后验稳健性。我们通过数值研究检验了基于完整数据的后验分布与基于剔除异常值后的数据所得到的后验分布之间的库尔贝克-莱布勒散度。