Heavy-tailed models are often used as a way to gain robustness against outliers in Bayesian analyses. On the other side, in frequentist analyses, M-estimators are often employed. In this paper, the two approaches are reconciled by considering M-estimators as maximum likelihood estimators of heavy-tailed models. We realize that, even from this perspective, there is a fundamental difference in that frequentists do not require these heavy-tailed models to be proper. It is shown what the difference between improper and proper heavy-tailed models can be in terms of estimation results through two real-data analyses based on linear regression. The findings of this paper make us ponder on the use of improper heavy-tailed data models in Bayesian analyses, an approach which is seen to fit within the generalized Bayesian framework of Bissiri et al. (2016) when combined with proper prior distributions yielding proper (generalized) posterior distributions.

翻译：在贝叶斯分析中，重尾模型常被用作获取对异常值稳健性的手段。另一方面，在频率主义分析中，M估计量则被广泛采用。本文通过将M估计量视为重尾模型的最大似然估计量，实现了两种方法的统一。我们认识到，即使从这个视角出发，两者仍存在根本差异：频率主义者并不要求这些重尾模型必须是正规的。通过基于线性回归的两个实际数据分析，本文展示了非正规与正规重尾模型在估计结果层面可能产生的差异。本文的研究促使我们重新思考非正规重尾数据模型在贝叶斯分析中的应用——当结合能产生正规（广义）后验分布的正规先验分布时，该方法可纳入Bissiri等人（2016）提出的广义贝叶斯框架之中。