This book surveys results in the area sometimes denoted as "partial exchangeability" of "de Finetti type theorems". It is to be seen as an attempt to give sense to the general idea that there is a strong coupling between a statistical model and the statistical analysis. So strong that there is a canonical mathematical construction leading from the analysis to the model. Special sections are devoted to the study of sufficiency, of triviality of tails of Markov chains, studied e.g. by coupling methods, Martin boundaries and projective limits of Markov kernels and Polish spaces. In addition, many examples of extreme point models are treated in detail. This book is intended for researchers and graduate students in mathematical statistics and probability.