Exchangeability and sets of desirable gambles

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.

💡 Research Summary

The paper presents a comprehensive study of sets of desirable gambles (SDGs) as a flexible framework for modelling uncertainty, and investigates how the classical notion of exchangeability can be incorporated into this framework. After recalling the basic rationality axioms for SDGs—acceptance of non‑negative gambles, positive scaling, additivity, and upward closure—the authors show that any coherent SDG forms an upward‑closed convex cone in a finite‑dimensional gamble space. This geometric picture underlies the connection with linear previsions and traditional probability measures.

The core contribution lies in the analysis of exchangeable assessments. For a finite sequence of n random variables, exchangeability forces the desirability of a gamble to depend only on the count vector (the number of times each outcome occurs) rather than on the order of observations. Consequently, the set of exchangeable desirable gambles can be represented as a polytope in the space of count vectors. By normalising the count vector, one obtains a frequency vector that lives in the unit simplex. The authors prove that this simplex representation is in one‑to‑one correspondence with the original polytope, providing a clean geometric interpretation of exchangeability.

A second, deeper representation uses multivariate Bernstein (or Bernstein‑basis) polynomials. The paper demonstrates that any exchangeable SDG can be expressed as a positive linear combination of Bernstein polynomials evaluated at the frequency vector. This mirrors de Finetti’s representation theorem: just as an exchangeable probability distribution can be written as a mixture of i.i.d. Bernoulli laws, an exchangeable SDG can be written as a mixture of “Bernstein‑type” desirable gambles. The coefficients of the mixture are non‑negative and sum to one, ensuring that the resulting cone remains coherent.

The authors then extend the finite‑dimensional results to infinite sequences. By invoking a limit argument on the Bernstein representation, they prove an infinite‑exchangeability theorem for SDGs: an infinite exchangeable SDG is precisely a mixture of i.i.d. desirable gambles, where the mixing measure lives on the simplex of frequency vectors. This result is a direct analogue of de Finetti’s classic theorem, but it holds in the more general setting of desirability rather than probability alone.

A substantial part of the paper is devoted to updating and inference under exchangeability. When new data are observed, the natural extension (the most conservative coherent extension) of the original SDG is computed. The authors show that, under exchangeability, the natural extension preserves the exchangeable structure: the updated SDG remains a convex cone that can still be described by a count‑vector polytope or, equivalently, by a revised set of Bernstein coefficients. This guarantees that posterior inference respects the original symmetry assumptions without requiring ad‑hoc adjustments.

Finally, the paper tackles the problem of extendibility: under what conditions can a finite exchangeable SDG be extended to a longer or infinite exchangeable sequence? The answer is given in terms of consistency of the count‑vector polytope and the convergence of the associated Bernstein mixtures. If the mixture coefficients converge to a proper probability measure on the simplex, the finite model admits an infinite exchangeable extension. The authors provide necessary and sufficient conditions, illustrate them with examples, and discuss how these conditions relate to classical results on extendibility of exchangeable probability models.

Overall, the work bridges three perspectives—geometric (polytopes in count‑vector space), algebraic (convex cones of desirable gambles), and analytic (Bernstein polynomial mixtures)—to deliver a unified theory of exchangeable uncertainty modelling. It not only generalises de Finetti’s representation theorems to the setting of desirability but also supplies practical tools for conservative inference, model updating, and the construction of extendible exchangeable sequences, thereby opening new avenues for robust decision‑making under uncertainty in statistics, artificial intelligence, and related fields.