A study of relationships between confidence regions being Bayesian, and the existence of some generalizations of Fisher's notion of relevant subsets. For a betting scheme introduced by Buehler, and for finite parameter space, it is shown that non-Bayesian procedures allow a winning strategy for a statistician's adversary. It is further shown, for finite parameter space, non-Bayesian procedures must admit conditional confidence levels bounded away from the unconditional level, the converse to a theorem of Wallace. For general parameter space these results follow from a procedure not being weak Bayes in a certain sense.
"On Some Difficulties in a Frequency Theory of Inference." Ann. Statist. 1 (2) 241 - 250, March, 1973. https://doi.org/10.1214/aos/1176342362