My question was always the practical implication of violating the assumption when fitting a model. In revising a recent paper, I must address the request from a reviewer to explain the exchangeability assumption (which was initially buried in a citation). The second paragraph in Section 2.2 of the paper was added. I came to a better understanding of the assumption as a result of revising the paper. Not to brush aside something that I can't give a satisfactory explanation is obviously beneficial.
The exchangeability assumption is a generalization of the most commonly used assumption in classical statistics: i.i.d. The assumption that observations were independent identically distributed random variables makes the formulation of a likelihood function a simple and straightforward process, that is, we can simply multiply the densities of individual observations to form the likelihood function. Checking the conformity with the assumption is, however, another issue. The assumptions (both i.i.d. and exchangeability) are associated with the model we propose. When we fit a regression model we always check the residuals to see if they are i.i.d. normal with mean 0 and a constant variance. Non-conformity with the assumption often suggests a wrong model was used. The i.i.d. assumption is verified after a model is fit, usually with graphs. How about the assumption that observations are exchangeable? I assume that we should check for the assumption
