Representation theorems for partially exchangeable random variables

We provide representation theorems for both finite and countable sequences of finite-valued random variables that are considered to be partially exchangeable. In their most general form, our results are presented in terms of sets of desirable gambles, a very general framework for modelling uncertainty. Its key advantages are that it allows for imprecision, is more expressive than almost every other imprecise-probabilistic framework and makes conditioning on events with (lower) probability zero non-problematic. We translate our results to more conventional, although less general frameworks as well: lower previsions, linear previsions and probability measures. The usual, precise-probabilistic representation theorems for partially exchangeable random variables are obtained as special cases.