Why can’t we stop referring to Fisher’s post data p-values from a particular data set as a “probability”, when they are also correctly referred to as percentiles (under an assumed model)?
The probability interpretation is relevant before the experiment, if one were doing an honest Neyman-Pearson design, where \alpha is traded off against \beta for the experiment at hand. In that case, it is preferable to think in terms of Z scores, rather than probabilities. This hypothetical “probability” is arguably meaningless if one doesn’t plan to repeat the experiment.
For the justification of thinking about p-values in terms of Z scores, see:
Kulinskaya, Staudte, Morgenthaler (2008) Meta Analysis: A Guide to Calibrating and Combining Statistical Evidence p. xiv
Raymond Hubbard & M. J Bayarri (2003) Confusion Over Measures of Evidence (p’s) Versus Errors (α’s) in Classical Statistical Testing, The American Statistician, 57:3, 171-178, DOI: 10.1198/0003130031856
This paper by @Sander is also worth reading.
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