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# Towards a strictly frequentist theory of imprecise probability

*Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications*, PMLR 215:230-240, 2023.

#### Abstract

Strict frequentism defines probability as the limiting relative frequency in an infinite sequence. What if the limit does not exist? We present a broader theory, which is applicable also to statistical phenomena that exhibit diverging relative frequencies. In doing so, we develop a close connection with imprecise probability: the cluster points of relative frequencies yield a coherent upper prevision. We show that a natural frequentist definition of conditional probability recovers the generalized Bayes rule. We prove constructively that, for a finite set of elementary events, there exists a sequence for which the cluster points of relative frequencies coincide with a prespecified set, thereby providing strictly frequentist semantics for coherent upper previsions.