A confidence level is sometimes treated as a probability and accordingly given substantial "confidence." And with certain statistical models a confidence level can also be an objective posterior probability (Fraser and MacKay, 1975). An instance involving such a probability with a binary error was discussed at the Symposium on Foundations of Statistical Inference, University of Waterloo, 1970 (Godambe and Sprott, 1971, page 49). An example by Buehler (ibid., page 337) of a betting strategy and a generalization by Rubin (ibid., page 340) were subsequently presented to support a claim against the objective posterior probability and against too much "confidence." The ordinary Student confidence interval in appropriate contexts is also an objective posterior probability interval. An example by Buehler and Feddersen (1963) has been cited frequently as evidence against the validity of the objective posterior probability and against too much "confidence." This note censures the common procedure of assessment in terms of betting strategies and introduces a modified balanced procedure for such betting assessments. When assessed by this balanced procedure the Buehler and Buehler-Feddersen strategies are faced with large losses and thus do not support claims against the objective posteriors and confidence levels.
"Confidence, Posterior Probability, and the Buehler Example." Ann. Statist. 5 (5) 892 - 898, September, 1977. https://doi.org/10.1214/aos/1176343945