In their seminal 1990 paper, Wasserman and Kadane establish an upper bound for the Bayes' posterior probability of a measurable set $A$, when the prior lies in a class of probability measures $\mathcal{P}$ and the likelihood is precise. They also give a sufficient condition for such upper bound to hold with equality. In this paper, we introduce a generalization of their result by additionally addressing uncertainty related to the likelihood. We give an upper bound for the posterior probability when both the prior and the likelihood belong to a set of probabilities. Furthermore, we give a sufficient condition for this upper bound to become an equality. This result is interesting on its own, and has the potential of being applied to various fields of engineering (e.g. model predictive control), machine learning, and artificial intelligence.

翻译：在1990年的开创性论文中，Wasserman与Kadane建立了当先验属于概率测度类$\mathcal{P}$且似然函数精确时，可测集$A$的贝叶斯后验概率的上界，并给出了该上界成立等式的充分条件。本文通过进一步处理与似然函数相关的不确定性，推广了他们的结果。我们给出了当先验与似然函数均属于某一概率集时的后验概率上界，并提出了使该上界成为等式的充分条件。此结果不仅具有独立理论价值，更有望应用于工程控制（如模型预测控制）、机器学习及人工智能等多个领域。