Introduction to Imprecise Probabilities

3.4 Exchangeability

To give a further illustration of why invariance is important, we now focus on a particular instance of strongly invariant models: we consider a joint model for some random variables and we assume that this joint model is strongly invariant under permutations of them, so it represents the belief that the order of the variables does not matter. Such models are called exchangeable, and they were first studied in the precise case by Bruno de Finetti [216]. Exchangeability was later extended to the theory of coherent lower previsions by Walley [672, § 9.5]. Here, we follow the more detailed and extensive treatment by De Cooman et al. [213, 214].

By virtue of de Finetti's Representation Theorem [216], an exchangeable model can be seen as a convex mixture of multinomial models. This has given some ground [177, 216, 218] to the claim that aleatory probabilities and IID processes can be eliminated from statistics, and that we can restrict ourselves to considering exchangeable sequences instead.⁶

Consider random variables , …, taking values in the same non-empty and finite set . A subject's beliefs about the values that these random variables assume jointly in is given ...