We introduce the notion of worst-case posterior and worst-case likelihood sensitivity. These measure, respectively, the sensitivity of the expected cost to worst-case perturbations of the posterior distribution and worst-case perturbations of the likelihood of a Bayesian model. Each defines a quantitative measure of robustness. A decision maker concerned about the sensitivity of the out-of-sample expected cost to deviations from her assumptions will want a decision for which both sensitivities are small. We derive posterior and likelihood sensitivities for uncertainty sets defined in terms of deviation measures. Posterior sensitivity vanishes when the posterior variance shrinks to zero, which occurs when parameter uncertainty is eliminated from learning. Parameter learning does not eliminate likelihood sensitivity. A distributionally robust formulation of a Bayesian optimization problem makes a near-Pareto-optimal tradeoff between performance (expected cost) and robustness (posterior and likelihood sensitivity).

翻译：我们提出了最坏情况后验敏感性和最坏情况似然敏感性的概念。分别衡量了期望成本对后验分布最坏情况扰动，以及贝叶斯模型似然函数最坏情况扰动的敏感性。每种敏感性均构成了鲁棒性的定量度量。若决策者关注样本外期望成本对其假设偏差的敏感性，其将倾向于选择两种敏感性均较小的决策方案。我们推导了由偏差度量定义的不确定集下的后验敏感性与似然敏感性。当后验方差趋近于零时——即参数不确定性通过学习被消除时——后验敏感性消失。参数学习无法消除似然敏感性。贝叶斯优化问题的分布鲁棒公式可在性能（期望成本）与鲁棒性（后验敏感性和似然敏感性）之间实现近似帕累托最优的权衡。