Exchangeability and sets of desirable gambles

TL;DR

This paper explores the mathematical structure of sets of desirable gambles, providing exchangeability criteria, representation theorems, and geometric interpretations, advancing the understanding of uncertainty models in decision theory.

Contribution

It extends de Finetti's exchangeability theorems to sets of desirable gambles, linking geometric and polynomial representations, and discusses inference under exchangeability.

Findings

Finite representation via count vectors has a clear geometric interpretation.

Frequency vector representation relates to multivariate Bernstein polynomials.

The paper clarifies the relationship between different representations of exchangeable models.

Abstract

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti's finite and infinite representation theorems. We show that the finite representation in terms of count vectors has a very nice geometrical interpretation, and that the representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability and the extension of exchangeable sequences.