Strong validity, consonance, and conformal prediction

Valid prediction of future observations is an important and challenging problem. The two general approaches for quantifying uncertainty about the future value employ prediction regions and predictive distribution, respectively, with the latter usually considered to be more informative because it performs other prediction-related tasks. Standard notions of validity focus on the former, i.e., coverage probability bounds for prediction regions, but a notion of validity relevant to the other prediction-related tasks performed by the latter is lacking. In this paper, we present a new notion---strong prediction validity---relevant to these more general prediction tasks. We show that strong validity is connected to more familiar notions of coherence, and argue that imprecise probability considerations are required in order to achieve it. We go on to show that strong prediction validity can be achieved by interpreting the conformal prediction output as the contour function of a consonant plausibility function. We also offer an alternative characterization, based on a new nonparametric inferential model construction, wherein the appearance of consonance is more natural, and prove strong prediction validity.