Towards a Chaotic Probability Model for Frequentist Probability: The Univariate Case

We adopt the same mathematical model of a set M of probability measures as is central to the theory of coherent imprecise probability. However, we endow this model with an objective, frequentist interpretation in place of a behavioral subjective one. We seek to use M to model stable physical sources of time series data that have highly irregular behavior and not to model states of belief or knowledge that are assuredly imprecise. The approach we present in this paper is to understand a set of measures model M not as a traditional compound hypothesis, in which one of the measures in M is a true description, but rather as one in which none of the individual measures in M provides an adequate description of the potential behavior of the physical source as actualized in the form of a long time series. We provide an instrumental interpretation of random process measures consistent with M and the highly irregular physical phenomena we intend to model by M. This construction provides us with the basic tools for simulation of our models. We present a method to estimate M from data which studies any given data sequence by analyzing it into subsequences selected by a set of computable rules. We prove results that help us to choose an adequate set of rules and evaluate the performance of the estimator.