are there examples where the frequentist confidence interval is clearly superior to the Bayesian credible interval (as per the challenge implicitly made by Jaynes).
Here is an example: the true $\theta$ equals $10$ but the prior on $\theta$ is concentrated about $1$. I am doing statistics for a clinical trial, and $\theta$ measures the risk to death, so the Bayesian result is a disaster, isn't it ? More seriously, what is "the" Bayesian credible interval ? In other words: what is the selected prior ? Maybe Jaynes proposed an automatic way to select a prior, I don't know !
Bernardo proposed a "reference prior" to be used as a standard for scientific communication [and even a "reference credible interval" (Bernardo - objective credible regions)]. Assuming this is "the" Bayesian approach, now the question is: when is an interval superior to another one ? The frequentist properties of the Bayesian interval are not always optimal, but neither are the Bayesian properties of "the" frequentist interval
(by the way, what is "the" frequentist interval ? )
