Exchangeable lower previsions

This paper deals with belief models for both finite and countable sequences of exchangeable random variables taking a finite number of values. When such sequences of random variables are assumed to be exchangeable, this more-or-less means that the specific order in which they are observed is deemed irrelevant. The first detailed study of exchangeability was made by de Finetti [5] (with the terminology of ‘equivalent’ events). He proved the now famous representation theorem, which is often interpreted as stating that a sequence of random variables is exchangeable if it is conditionally independent and identically distributed (i.i.d.). Other important work on exchangeability was done by, amongst many others, Hewitt and Savage [12], Heath and Sudderth [10], Diaconis and Freedman [8] and, in the context of the behavioural theory of imprecise probabilities that we are going to consider here, by Walley [19]. We refer to Kallenberg [14, 15] for modern, measure-theoretic discussions of exchangeability. One of the reasons why exchangeability is deemed important, especially by Bayesians, is that, by virtue of de Finetti’s representation theorem, an exchangeable model can be seen as a convex mixture of multinomial models. This has lent some support [2, 5, 7] to the claim that aleatory probabilities and i.i.d. processes can be eliminated from statistics