In this paper, we demonstrate that 100(1-?)% frequentist confidence sets enjoy at least 100(1-?)% Bayesian credibility under a novel constraint termed "the autonomy principle". The principle stipulates that one cannot discriminate between samples in which a confidence set contains the relevant parameter and samples in which it does not. The result is obtained by pursuing a Bayesian interpretation of suitably restricted frequentist gambling behavior. We further show that, under the autonomy principle, frequentist confidence measures constitute Bayesian posterior measures. We contend that the principle often holds in the absence of prior information, and we discuss its connection with objective Bayesian inference.