Expressing Regret: A Unified View of Credible Intervals

Posterior uncertainty is typically summarized as a credible interval, an interval in the parameter space that contains a fixed proportion—usually 95%—of the posterior’s support. For multivariate parameters, credible sets perform the same role. There are of course many potential 95% intervals from which to choose, yet even standard choices are rarely justified in any formal way. In this article we give a general method, focusing on the loss function that motivates an estimate—the Bayes rule—around which we construct a credible set. The set contains all points which, as estimates, would have minimally-worse expected loss than the Bayes rule: we call this excess expected loss “regret.” The approach can be used for any model and prior, and we show how it justifies all widely used choices of credible interval/set. Further examples show how it provides insights into more complex estimation problems. Supplementary materials for this article are available online.

后验不确定性通常以可信区间（credible interval）进行概括，可信区间是参数空间内包含固定比例（通常为95%）后验分布支撑集的区间。对于多变量参数，可信集（credible set）可发挥同等作用。当然存在诸多可供选择的95%区间，但即便主流选用的区间，也极少能通过任何形式化方式证明其合理性。本文提出一种通用方法，以驱动估计量的损失函数——即贝叶斯准则（Bayes rule）——为核心构建可信集。该可信集包含所有作为估计量时，其期望损失仅略高于贝叶斯准则的点：我们将这部分额外的期望损失称为“遗憾（regret）”。本方法可适用于任意模型与先验分布，且本文将证明其可对所有广泛使用的可信区间/集选择给出合理性解释。更多示例则展示了该方法如何为更复杂的估计问题提供分析视角。本文的补充材料可在线获取。