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.

对未来观测的预测是一个重要且具有挑战性的问题。量化预测不确定性的两种主流方法分别使用预测区域和预测分布，后者被认为提供更多信息，因为它可以执行其他与预测相关的任务。有效性的标准概念，我们在这里称为类型 1 有效性，侧重于预测区域的覆盖概率，而缺乏与预测分布执行的其他预测相关任务相关的有效性概念。在这里，我们提出了一个新概念，称为Type-2 有效性，与这些其他预测任务相关。我们建立了类型 2 有效性和连贯性之间的联系，并表明为了实现它需要不精确的概率考虑。我们继续表明，通过将共形预测输出解释为辅音似真度量的轮廓函数，可以实现两种类型的预测有效性。我们还提供了基于新的非参数推理模型构造的共形预测的替代表征，其中协和的出现是自然的，并证明其有效性。