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:Robust Bayust Bayesian 线性回归是一个古典但重要的统计工具。 虽然最近在一个错误分布类别下证明了后方分布的新强度特性, 但其充分条件是限制性的, 并排除了几个重要情况。 在这项工作中, 我们重新审视了典型的两种成分的应对变量模型, 也称为污染模型, 其中一种成分是轻尾回归模型, 另一成分是重尾。 后一种成分独立于回归参数, 这对于证明后方强度至关重要。 我们通过使用这些条件获得了后方( 非) 的充足新条件, 并揭示了非三角强度结果 。 特别是, 我们发现即使是学生- $t 错误分布也能够实现我们框架中的后方强度。 进行了一项数字研究, 以检查基于完整数据的后方分布和基于删除外源获得的数据之间的 Kullback- Leiber 差异 。</s>