贝叶斯分布鲁棒优化中的后验和似然敏感性
Posterior and Likelihood Sensitivity in Bayesian Distributionally Robust Optimization
AI总结：
本文提出最坏情况后验和似然敏感性的概念，用于量化贝叶斯模型对后验和似然扰动的鲁棒性，并证明分布鲁棒优化可实现性能与鲁棒性的近似帕累托最优权衡。
摘要：
我们引入了最坏情况后验和似然敏感性的概念。这些分别衡量期望成本对后验分布最坏情况扰动和贝叶斯模型似然最坏情况扰动的敏感性。每个都定义了鲁棒性的定量度量。关心样本外期望成本对其假设偏差敏感性的决策者将希望两个敏感性都较小的决策。我们推导了由偏差度量定义的不确定性集的后验和似然敏感性。当后验方差缩小到零时，后验敏感性消失，这发生在参数不确定性通过学习消除时。参数学习不能消除似然敏感性。贝叶斯优化问题的分布鲁棒公式在性能（期望成本）和鲁棒性（后验和似然敏感性）之间实现了近似帕累托最优的权衡。
英文摘要：
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).
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