Independent Natural Extension for Infinite Spaces

TL;DR

This paper introduces a new, always-existing notion of independent natural extension for infinite spaces within uncertainty models, using Williams-coherence, and compares it to previous approaches, highlighting its properties and special cases.

Contribution

It proposes a novel independent natural extension framework for infinite spaces that always exists and satisfies key properties, contrasting with prior models that lack these guarantees.

Findings

The independent natural extension always exists under Williams-coherence.

It satisfies properties like factorisation and external additivity.

Epistemic independence includes conventional independence as a special case.

Abstract

We define and study the independent natural extension of two local uncertainty models for the general case of infinite spaces, using the frameworks of sets of desirable gambles and conditional lower previsions. In contrast to Miranda and Zaffalon (2015), we adopt Williams-coherence instead of Walley-coherence. We show that our notion of independent natural extension always exists - whereas theirs does not - and that it satisfies various convenient properties, including factorisation and external additivity. The strength of these properties depends on the specific type of epistemic independence that is adopted. In particular, epistemic event-independence is shown to outperform epistemic atom-independence. Finally, the cases of lower expectations, expectations, lower probabilities and probabilities are obtained as special instances of our general definition. By applying our results to these instances, we demonstrate that epistemic independence is indeed epistemic, and that it includes the conventional notion of independence as a special case.