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 ...

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