I am under the impression that there must be a duality between physical theories and statistical models that have no reference to physical theories. For instance, given an apparatus, a researcher can begin to collect data and form statistical models for expected results given preparations. I believe this is called operationalism, or perhaps instrumentalism. Suppose you have a successful physical theory and also an instrumental statistical model that predicts experiments, what is the relationship (precisely speaking) that relates the two? What are the main references that attempt to address this? There has been considerable work on this subject in quantum mechanics. What about classical mechanics?