A Conditional Confidence Principle

A conditional confidence property is examined in the context of invariant statistical models, for set estimators of equivariant functions of the parameter. Set estimators deduced from likelihood considerations are then identical to Bayes credible sets induced from a right invariant prior. It is shown that amenability of the group ensures that these intervals satisfy a betting interpretation of confidence sets due essentially to Buehler and Tukey. As a corollary, a level α Bayes set estimator is of level at-most-α as a Neyman-Pearson confidence interval if the group is amenable.