Prediction of future observations is an important and challenging problem. The two mainstream approaches for quantifying prediction uncertainty use prediction regions and predictive distributions, respectively, with the latter believed to be more informative because it can perform other prediction-related tasks. The standard notion of validity, what we refer to here as Type-1 validity, focuses on coverage probability of prediction regions, while a notion of validity relevant to the other prediction-related tasks performed by predictive distributions is lacking. Here we present a new notion, called Type-2 validity, relevant to these other prediction tasks. We establish connections between Type-2 validity and coherence properties, and show that imprecise probability considerations are required in order to achieve it. We go on to show that both types of prediction validity can be achieved by interpreting the conformal prediction output as the contour function of a consonant plausibility measure. We also offer an alternative characterization of conformal prediction, based on a new nonparametric inferential model construction, wherein the appearance of consonance is natural, and prove its validity.
翻译:预测未来观测的预测是一个重要而具有挑战性的问题。两种量化预测不确定性的主流方法分别使用预测区域和预测分布,预测分布被认为信息更加丰富,因为它能够执行其他预测相关的任务。标准的有效性概念,我们这里称之为类型1的有效性,侧重于预测区域的覆盖概率,而缺乏与预测分布所执行的其他预测相关任务相关的有效性概念。我们在这里提出了一个新概念,称为类型2有效性,与这些其他预测任务相关。我们在类型2的有效性和一致性特性之间建立了联系,并表明为实现这一特性需要精确的概率考虑。我们继续表明,将符合的预测产出解释为相容的相容性计量的轮廓功能,可以实现两种预测的有效性。我们还根据新的非参数性推论模型的构造,对符合的预测作了另外的定性,即共性是自然的,并证明其有效性。