Strictly Frequentist Imprecise Probability

TL;DR

This paper extends strict frequentist probability to cases where relative frequencies diverge, connecting it with imprecise probability and providing a generalized, coherent framework for such phenomena.

Contribution

It introduces a broader frequentist theory applicable to diverging relative frequencies, linking it with imprecise probability and establishing new concepts of conditional probability and independence.

Findings

Cluster points of relative frequencies form a coherent upper prevision.

Generalized Bayes rule is recovered under the new framework.

Existence of sequences matching prescribed cluster points demonstrates naturalness.

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 random phenomena that exhibit diverging relative frequencies. In doing so, we develop a close connection with the theory of 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. This also suggests an independence concept, which is related to epistemic irrelevance in the imprecise probability literature. Finally, 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 which demonstrates the naturalness, and arguably completeness, of our theory.