Abstract

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.

摘要

后验不确定性通常被概括为可信区间，参数空间中的区间包含固定比例（通常为 95%）的后验支持。对于多变量参数，可信集起着同样的作用。当然，有许多潜在的 95% 区间可供选择，但即使是标准选择也很少以任何形式证明是合理的。在这篇文章中，我们给出了一种通用方法，重点关注激发估计的损失函数——贝叶斯法则——我们围绕它构建了一个可信集。该集合包含所有点，据估计，这些点的预期损失会比贝叶斯规则差得最少：我们称这种超额预期损失为“遗憾”。该方法可用于任何模型和先验，我们展示了它如何证明所有广泛使用的可信区间/集合选择的合理性。进一步的例子展示了它如何提供对更复杂的估计问题的洞察力。本文的补充材料可在线获取。