We provide a theoretical framework for a wide class of generalized posteriors that can be viewed as the natural Bayesian posterior counterpart of the class of M-estimators in the frequentist world. We call the members of this class M-posteriors and show that they are asymptotically normally distributed under mild conditions on the M-estimation loss and the prior. In particular, an M-posterior contracts in probability around a normal distribution centered at an M-estimator, showing frequentist consistency and suggesting some degree of robustness depending on the reference M-estimator. We formalize the robustness properties of the M-posteriors by a new characterization of the posterior influence function and a novel definition of breakdown point adapted for posterior distributions. We illustrate the wide applicability of our theory in various popular models and illustrate their empirical relevance in some numerical examples.

翻译：我们为一大类广义后验提供了一个理论框架，这类广义后验可视为频率派领域中M-估计量类在贝叶斯后验中的自然对应物。我们将该类的成员称为M-后验，并证明在M-估计损失函数和先验分布的温和条件下，它们是渐近正态分布的。具体而言，M-后验在概率意义上收缩于一个以M-估计量为中心的正态分布周围，这显示了频率派的一致性，并暗示了依赖于参考M-估计量的某种程度的稳健性。我们通过后验影响函数的新刻画以及适用于后验分布的新颖崩溃点定义，形式化了M-后验的稳健性性质。我们在多种流行模型中阐述了我们理论的广泛适用性，并通过一些数值示例说明了其经验相关性。