I was reminded of this old post by Andrew Gelman about whether Quantum Mechanics requires a change in the axioms of probability theory. Without weighing in I’ll just point out that Bayesian Statistics is more general than Bayes Theorem and this affects the controversy.
With some poetic license we can say Bayesians take the axioms of probability theory (sum and product rule) as fundamental and we’d rather solve every problem with them than invent new ad-hoc principles like p-values. So when it comes to judging hypothesis x based on data D we use the product rule to derive Bayes Theorem:
But this only applies in certain contexts, or at least, it’s only usable in certain contexts. Under different conditions the axioms imply quite different formulas. If for example there is a nuisance parameter they lead to this updating formula:
If D only affects x through an intermediary then they imply the correct updating formula is:
If x is determined by averaging over models then they are related by the following system of equations:
The particular examples aren’t important. The point is that there are an infinite number of contexts in statistics and so the axioms imply an infinite number of “updating formulas”. Bayes Theorem is merely the simplest.
So every researcher has two choices. They can believe we understand the context in QM and therefore need a weird new form of probability theory. Or they can keep the old axioms and search for the context which generates the peculiar formulas seen in QM.
Which research path is chosen seems to depend largely on philosophy and temperament. Everyone has to make their choice and live with the consequences though, since it’s doubtful they’re equally fruitful. Sometimes philosophy really does make a difference.