On exchangeable sampling distributions for uncontrolled data

When statistical observations are not based upon a controlled randomized experiment, it can be appealing to try to model their joint distribution via an exchangeable sampling distribution. However, exchangeable sampling distributions should be used with extreme caution, and do not obviously usefully model any lack of independence of the observation vectors. The two main problems concern the distributions of the test statistics, together with a lack of identifiability of the dependencies between the observation vectors. Two new asymptotic results relating to empirical processes, Dirichlet processes, and non-parametric tests for fit, are described in order to highlight these problems.