Mark Palko points us to this 2009 article by Itzhak Gilboa, Andrew Postlewaite, and David Schmeidler, which begins:
This note argues that, under some circumstances, it is more rational not to behave in accordance with a Bayesian prior than to do so. The starting point is that in the absence of information, choosing a prior is arbitrary. If the prior is to have meaningful implications, it is more rational to admit that one does not have sufficient information to generate a prior than to pretend that one does. This suggests a view of rationality that requires a compromise between internal coherence and justification, similarly to compromises that appear in moral dilemmas. Finally, it is argued that Savage’s axioms are more compelling when applied to a naturally given state space than to an analytically constructed one; in the latter case, it may be more rational to violate the axioms than to be Bayesian.
The paper expresses various misconceptions, for example the statement that the Bayesian approach requires a “subjective belief.” All statistical conclusions require assumptions, and a Bayesian prior distribution can be as subjective or un-subjective as any other assumption in the model. For example, I don’t recall seeing textbooks on statistical methods referring to the subjective belief underlying logistic regression or the Poisson distribution; I guess if you assume a model but you don’t use the word “Bayes,” then assumptions are just assumptions.
More generally, it seems obvious to me that no statistical method will work best under all circumstances, hence I have no disagreement whatsoever with the opening sentence quoted above. I can’t quite see why they need 12 pages to make this argument, but whatever.
P.S. Also relevant is this discussion from a few years ago: The fallacy of the excluded middle—statistical philosophy edition.