> Correct, which is why it's crazy to consider a Jeffrey's credible interval as valid, but not a confidence interval.
The confidence interval is “valid” for what its definition is. But:
- saying “there is a 95% chance that this interval contains the true value because it’s a 95% credible interval” is valid;
- saying “there is a 95% chance that this interval contains the true value because it’s a 95% confidence interval” is not.
It’s very much possible to reach a correct conclusion with invalid reasoning. As Jaynes put it:
“As we stress repeatedly, from a false proposition – or from a fallacious argument that leads to a false proposition – all propositions, true and false, may be deduced. But this is just the danger; if fallacious reasoning always led to absurd conclusions, it would be found out at once and corrected. But once an easy, shortcut mode of reasoning has led to a few correct results, almost everybody accepts it; those who try to warn against it are not listened to.”
> Perhaps the confidence interval might not be valid under a frequentist interpretation, but the same math can be valid under a Bayesian interpretation even if the person publishing didn't mean it that way.
Right, but if you get that by happenstance and not from the fact that it’s a confidence interval, the latter isn’t really getting you anything.
Per my first link: “One of the misconceptions regarding the relationship between Bayesian inference and frequentist inference is that they will lead to the same inferences, and hence all confidence intervals can simply be interpreted in a Bayesian way. In the case where data are normally distributed, for instance, there is a particular prior that will lead to a confidence interval that is numerically identical to Bayesian credible intervals computed using the Bayesian posterior (Jeffreys 1961; Lindley 1965). This might lead one to suspect that it does not matter whether one uses confidence procedures or Bayesian procedures. We showed, however, that confidence intervals and credible intervals can disagree markedly. The only way to know that a confidence interval is numerically identical to some credible interval is to prove it. The correspondence cannot — and should not — be assumed.
More broadly, the defense of confidence procedures by noting that, in some restricted cases, they numerically correspond to Bayesian procedures is actually no defense at all. One must first choose which confidence procedure, of many, to use; if one is committed to the procedure that allows a Bayesian interpretation, then one’s time is much better spent simply applying Bayesian theory. If the benefits of Bayesian theory are desired — and they clearly are, by proponents of confidence intervals — then there is no reason why Bayesian inference should not be applied in its full generality, rather than using the occasional correspondence with credible intervals as a hand-waving defense of confidence intervals.”
> which is why the credible interval is not some magical better alternative - it's mostly just a choice of how you interpret it
You don’t get to choose to interpret a confidence interval as if it had the meaning of a Bayesian one. If you want to be able to interpret a computed interval as “there is a 95% chance that this interval contains the true parameter value”, only the Bayesian procedure guarantees it, and it can very well produce different numbers. So it is a better alternative in that sense.