Probability Models in Statistical Data Analysis: Uses, Interpretations, Frequentism-As-Model

Note: Accepted for publication as a chapter in "Handbook of the History and Philosophy of Mathematical Practice - Practical, Historical and Philosophical Instances of Probability" (Springer Nature, editor Egan Chernoff) The application of mathematical probability theory in statistics is quite controversial. Controversies regard both the interpretation of probability, and approaches to statistical inference. After having given an overview of the main approaches, I will propose a re-interpretation of frequentist probability. Most statisticians are aware that probability models interpreted in a frequentist manner are not really true in objective reality, but only idealisations. I argue that this is often ignored when actually applying frequentist methods and interpreting the results, and that keeping up the awareness for the essential difference between reality and models can lead to a more appropriate use and interpretation of frequentist models and methods, called "frequentism-as-model". This is elaborated showing connections to existing work, appreciating the special role of independently and identically distributed observations and subject matter knowledge, giving an account of how and under what conditions models that are not true can be useful, giving detailed interpretations of tests and confidence intervals, confronting their implicit compatibility logic with the inverse probability logic of Bayesian inference, re-interpreting the role of model assumptions, appreciating robustness, and the role of "interpretative equivalence" of models. Epistemic probability shares the issue that its models are only idealisations, and an analogous "epistemic-probability-as-model" can also be developed.