A Particular Upper Expectation as Global Belief Model for Discrete-Time Finite-State Uncertain Processes

TL;DR

This paper introduces a unique, axiomatic global belief model for discrete-time finite-state uncertain processes, aligning it with game-theoretic and measure-theoretic upper expectations, ensuring a conservative and consistent belief framework.

Contribution

It provides an axiomatic characterization of the game-theoretic upper expectation, establishing its uniqueness and relation to traditional measure-theoretic models.

Findings

The model coincides with Shafer and Vovk's game-theoretic upper expectation.

The measure-theoretic upper expectation satisfies the same axioms.

All models agree when local models are precise.

Abstract

To model discrete-time finite-state uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that is the most conservative one under a set of basic axioms. Our motivation for these axioms, which describe how local and global belief models should be related, is based on two possible interpretations for an upper expectation: a behavioural one similar to Walley's, and an interpretation in terms of upper envelopes of linear expectations. We show that the most conservative upper expectation satisfying our axioms, that is, our model of choice, coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications: it guarantees that there is a unique most conservative global belief model satisfying our axioms; and it shows that Shafer and Vovk's model can be given an axiomatic characterisation and thereby provides an alternative motivation for adopting this model, even outside their game-theoretic framework. Finally, we relate our model to the upper expectation resulting from a traditional measure-theoretic approach. We show that this measure-theoretic upper expectation also satisfies the proposed axioms, which implies that it is dominated by our model or, equivalently, the game-theoretic model. Moreover, if all local models are precise, all three models coincide.