Exchangeable lower previsions

TL;DR

This paper generalizes de Finetti's exchangeability concept to beliefs modeled by coherent lower previsions, providing representation theorems, convergence results, and methods for exchangeable natural extension.

Contribution

It introduces a framework for exchangeability under lower previsions, extending classical results and solving the exchangeable natural extension problem.

Findings

Representation theorems for finite and countable exchangeable sequences

Convergence of sample means in exchangeable sequences

Method for the most conservative exchangeable lower prevision

Abstract

We extend de Finetti's (1937) notion of exchangeability to finite and countable sequences of variables, when a subject's beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We prove representation theorems in both the finite and the countable case, in terms of sampling without and with replacement, respectively. We also establish a convergence result for sample means of exchangeable sequences. Finally, we study and solve the problem of exchangeable natural extension: how to find the most conservative (point-wise smallest) coherent and exchangeable lower prevision that dominates a given lower prevision.