Coverage Guarantees as Ambiguity Sets: Coherent Decision Making under Statistical Uncertainty

Confidence and conformal sets are used to inform decisions. To employ them, a decision maker must give it a probabilistic interpretation, and that interpretation determines an ambiguity set: the probability distributions over states she regards as possible. This paper characterizes when a proffered confidence/conformal set admits a coherent probabilistic interpretation, a property we call rationalizability. We then study decision making over a rationalizable confidence/conformal set. A common practice assigns zero probability to states outside the proffered set. This replaces the original coverage requirement with one that can become incompatible with additional probabilistic information. The rationalizable set, however, preserves the original guarantee and remains compatible with information refinements. As information accumulates and the ambiguity set shrinks toward a singleton, the associated robust decision rule converges to Bayesian expected-loss minimization. This consistency under information refinement is our normative justification for the coherent interpretation.