Robust Bayesian analysis has been mainly devoted to detecting and measuring robustness to the prior distribution. Indeed, many contributions in the literature aim to define suitable classes of priors which allow the computation of variations of quantities of interest while the prior changes within those classes. The literature has devoted much less attention to the robustness of Bayesian methods to the likelihood function due to mathematical and computational complexity, and because it is often arguably considered a more objective choice compared to the prior. In this contribution, a new approach to Bayesian local robustness to the likelihood function is proposed and extended to robustness to the prior and to both. This approach is based on the notion of distortion function introduced in the literature on risk theory, and then successfully adopted to build suitable classes of priors for Bayesian global robustness to the prior. The novel robustness measure is a local sensitivity measure that turns out to be very tractable and easy to compute for certain classes of distortion functions. Asymptotic properties are derived and numerical experiments illustrate the theory and its applicability for modelling purposes.

翻译：稳健贝叶斯分析主要致力于检测和度量对先验分布的稳健性。事实上，文献中的许多贡献旨在定义合适的先验类别，使得当先验在这些类别内变化时，能够计算感兴趣量的变化。由于数学和计算复杂性，且似然函数通常被认为比先验更具客观性，文献对贝叶斯方法在似然函数上的稳健性关注较少。本文提出了一种针对似然函数的贝叶斯局部稳健性新方法，并将其扩展至先验及两者的稳健性分析。该方法基于风险理论文献中引入的失真函数概念，该概念随后被成功用于构建适用于先验贝叶斯全局稳健性的先验类别。这一新颖的稳健性度量是一种局部敏感性度量，对于某些失真函数类别具有高度可处理性和易计算性。本文推导了渐近性质，并通过数值实验说明了该理论及其在建模目的中的适用性。