 This approach helps to clarify the role of the assumptions behind probabilistic models. One assumption that one can make about the observations is that they are “exchangeable”. This means that the joint distribution of the random variables is invariant under permutations. Informally, this means that the order of the observations does not affect the probability. For instance, in the urn case, this would mean that drawing first a white ball and then a black ball is just as probable as first drawing a black and then a white. De Finetti proved a general representation theorem that if the joint probability distribution of an infinite sequence of random variables is assumed to be exchangeable, then it can be written as a mixture of distribution functions from each of which the data behave as if they are independent random draws (de Finetti 1964). In the case of the urn example, the theorem shows that it is as if the data are independent random draws from a binomial distribution over a parameter \(\theta\), which itself has a prior probability distribution.