Statistical Compatibility, Refutational Information, and Acceptability

Abstract: This paper develops an interpretive framework for divergence P-values and S-values within a descriptive frequentist perspective. Statistical analysis is framed as operating within idealized worlds defined by a set of assumptions and a target hypothesis, where probabilities describe the behavior of data under the model but do not assign truth values to hypotheses. Within this view, P-values are interpreted as graded indices of compatibility between the observed result and the predictions generated by the assumed model; accordingly, small P-values should not be read as indicating logical impossibility or strict inconsistency of the model itself. Building on this distinction, the paper argues that practical inference requires moving beyond the internal logic of the model toward judgments of overall acceptability, which depend not only on data-model compatibility but also on multiple contextual considerations such as subject-matter knowledge, plausibility of assumptions, data quality, usefulness, and loss - all interpreted through the competence, intentions, perceptions, and moral values of the specific analyst. S-values are therefore interpreted not as evidence against the epistemic status of the model, but as a specific form of refutational information that contributes to the broader body of information used by the analyst to judge whether a model remains acceptable for an intended practical purpose. The paper also examines the linguistic and conceptual risks associated with the language of incompatibility, distinguishes probability from rarity, and clarifies different notions of surprise - including a possible definition of Shannon-type surprise, to be distinguished from Bayesian belief revision. Overall, the article proposes a more cautious and explicit interpretation of frequentist measures, centered on model-based description, analyst responsibility, and decision acceptability.