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TobyBartels comments on Bayes' rule =/= Bayesian inference - Less Wrong
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There's really no contradiction between the classical and Bayesian results:
They are saying different things.
If the 95% credible intervals are different, I think they are in contradiction.
The frequentist does not supply a credible interval, but a confidence interval. The credible interval has to do with the posterior P(H|D) (and thus the prior P(H)), which the frequentist refuses to talk about. The confidence interval has to do P(D|H) (with p-values and maybe likelihood ratios). Scientists often treat confidence intervals as credible intervals, and thus are wrong, but frequentist statisticians are more sophisticated. I'm not clear on what they say, though.
One advantage of the confidence interval is that it makes it clear that the test wasn't that powerful and that the bayesian is relying on the prior. The bayesian is not going to change the mind of the owner of the coin, who clearly has a different prior.
Incidentally, if someone claimed a coin produced 90% heads, I'd update away from such a sharply peaked distribution.