A central theme of Don Fraser's article, titled "Is Bayes Posterior just Quick and Dirty Confidence?", was that Bayesian confidence regions have approximate, and sometimes poor frequentist coverage (i.e., the frequency with which a confidence region contains the true parameter value under repeated sampling).
Fraser has this warning:
The failure to make true assertions with a promised reliability can be extreme with the Bayes use of mathematical priors (Stainforth et al., 2007; Heinrich, 2006). The claim of a probability status for a statement that can fail to be approximate confidence is misrepresentation. In other areas of science such false claims would be treated seriously.
The complaint about coverage of Bayesian confidence regions arises often, I think, because there is a very ingrained notion that correct frequentist coverage is a most desirable quality; that frequentist coverage is a literal statement about a natural phenomenon. Fraser goes further to say that confidence regions with incorrect coverage are misrepresentative, perhaps even fraudulent (i.e., a thing to be 'treated seriously')!
Of course, frequentist coverage is almost never a literal statement about a natural phenomenon, because statistical models almost never fully reflect the truth. In the sentiment of G.E.P. Box, all reported confidence levels are wrong, but some are useful.
More importantly, the criticism of approximate frequentist coverage is readily dismissed from a Bayesian perspective. In response, Christian Robert had this final comment:
Bayesian credible intervals are not frequentist confidence intervals and thus do not derive their optimality from providing an exact frequentist coverage.
On a side note, Fraser's statement above seems to neglect that frequentist probability is different from Bayesian probability in the context of confidence regions! I wrote on a related topic a few weeks ago: Bayesian vs. Frequentist Intervals: Which are more natural to scientists? Unfortunately though, my understanding is upset again due to Robert's reference to Jaynes: "there is only one kind of probability". How can this be true? Aren't Bayesians clear that the posterior distribution on a parameter is not to be interpreted in a frequentist way? And aren't frequentists clear that confidence regions not to be interpreted in a subjective way?