We propose a new type of statistical interval obtained by weakening the definition of a p% credible interval: After observing the interval, but not inspecting the full dataset ourselves, we should put at least a p% belief in it. From a decision-theoretical point of view the resulting intervals occupy a middle ground between frequentist and fully Bayesian statistical intervals, both practically and philosophically: To a p% Bayesian credible interval we should assign (at least a) p% belief after observing both the dataset and the interval, while p% frequentist intervals we can in general only assign a p% belief before seeing either the data or the interval. We derive concrete implementations for two cases: estimation of the fraction of a distribution that falls below a certain value (i.e., the CDF), and of the mean of a distribution with bounded support. Even though the problems are fully non-parametric, these methods require only one-dimensional priors. They share many of the practical advantages of Bayesian methods while avoiding the complexity of assigning high-dimensional priors altogether. Asymptotically they give intervals equivalent to the fully Bayesian approach and somewhat wider intervals, respectively. We discuss promising directions where the proposed type of interval may provide significant advantages.

翻译：本文提出一种通过弱化p%可信区间定义而获得的新型统计区间：在观察到该区间但未检查完整数据集的情况下，我们应至少对其持有p%的置信度。从决策理论的角度看，所得区间在实践与哲学层面均处于频率学派区间与完全贝叶斯统计区间的中间地带：对于p%贝叶斯可信区间，我们应在同时观测数据集与区间后赋予（至少）p%的置信度；而对于p%频率学派区间，通常我们仅能在未观测数据或区间前赋予p%的置信度。我们针对两种情况推导具体实现方法：估计分布低于特定值的比例（即累积分布函数），以及估计有界支撑集分布的平均值。尽管问题完全属于非参数范畴，这些方法仅需一维先验。它们在保留贝叶斯方法诸多实践优势的同时，完全避免了高维先验分配的复杂性。渐近分析表明，这两种方法分别产生与完全贝叶斯方法等效的区间及稍宽的区间。我们探讨了所提区间类型可能具有显著优势的研究方向。