We propose a new type of statistical interval obtained by weakening the definition of a p% credible interval: Having observed the interval (rather than the full dataset) 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 also after seeing the full dataset, while p% frequentist intervals we 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：我们提出一种通过弱化p%置信区间定义而获得的新型统计区间：在观测到区间（而非完整数据集）后，我们应至少对其赋予p%的置信度。从决策理论视角看，所得区间在实践与哲学层面均处于频率主义区间与完全贝叶斯统计区间之间的中间地带：对于p%贝叶斯置信区间，我们在看到完整数据集后仍应（至少）赋予其p%的置信度；而对于p%频率主义区间，我们通常仅在观测到数据或区间之前赋予其p%的置信度。我们针对两种情况推导出具体实现方法：估计分布中低于特定值的比例（即累积分布函数），以及估计有界支撑集分布的平均值。尽管这些问题完全是非参数化的，但所述方法仅需一维先验。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赋值的复杂性。渐近分析表明，这两种方法分别给出与完全贝叶斯方法等价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若干前景方向。