Ah, but which probability theory? Bayesian or frequentist? Or the ideas of Fisher?
How do you feel about the likelihood principle? The Behrens-Fisher problem, particularly when the variances are unknown and not assumed to be equal? The test of a sharp (or point) null hypothesis?
It does no good to assume that one’s statistics and probability theory are not built on axioms themselves. I have rarely met a probabilist or statistician whose answer about whether he or she believes in the likelihood principle or in the logically contradicted significance tests (or in various solutions of the Behrens-Fisher problem) does not depend on some sort of axiom or idea of what simply “seems right.” Of course, there are plenty of scientists who use mutually contradictory statistical tests, depending on what they’re doing.
Probability theory still applies.
Ah, but which probability theory? Bayesian or frequentist? Or the ideas of Fisher?
How do you feel about the likelihood principle? The Behrens-Fisher problem, particularly when the variances are unknown and not assumed to be equal? The test of a sharp (or point) null hypothesis?
It does no good to assume that one’s statistics and probability theory are not built on axioms themselves. I have rarely met a probabilist or statistician whose answer about whether he or she believes in the likelihood principle or in the logically contradicted significance tests (or in various solutions of the Behrens-Fisher problem) does not depend on some sort of axiom or idea of what simply “seems right.” Of course, there are plenty of scientists who use mutually contradictory statistical tests, depending on what they’re doing.