It is often the case that a confidence interval with 95% coverage is very similar to a credible interval that contains 95% of the posterior density. This happens when the prior is uniform or near uniform in the latter case. Thus a confidence interval can often be used to approximate a credible interval and vice versa. Importantly, we can conclude from this that the much maligned misinterpretation of a confidence interval as a credible interval has little to no practical importance for many simple use cases.
There are a number of examples out there of cases where this does not happen, however they all seem to be cherrypicked by proponents of Bayesian stats in an attempt to prove there is something wrong with the frequentist approach. In these examples, we see the confidence interval contains impossible values, etc which is supposed to show that they are nonsense.
I don’t want to go back over those examples, or a philosophical discussion of Bayesian vs Frequentist.
I am just looking for examples of the opposite. Are there any cases where the confidence and credible intervals are substantially different, and the interval provided by the confidence procedure is clearly superior?