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Okay, so, credible intervals aren't the same as confidence intervals. We all know that. In fact, they're only guaranteed to be the same when they're about a location or a scale parametre with a noninformative prior when they can be calculated from a single sufficient statistic. Furthermore, the most obvious way in which they differ is exactly in the prior, since if you have an informative prior then the credible interval even for the mean of a normal distribution will be different than the confidence one.

However, is there a simple example of a continuous case where they differ even with noninformative priors, that's not Jaynes' Truncated Exponential?

Googling finds the truncated exponential, also the other time this question was answered here but that's discrete and not terribly clear either.

So, what I want is some example of a continuous variable that's reasonably simple (by reasonably simple I mean "doesn't need obscure theorems like the characteristic function theorem") where the prior is noninformative and confidence and credible intervals do not coincide.

--EDIT:

I wonder why people marked this question as a duplicate of another question when I explicitly linked to that other question and said that it did not answer my question, and therefore those were not the answers I was looking for. Specifically because they were either too complex or too discrete.

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  • $\begingroup$ You might look at this paper which has a few examples where they differ. $\endgroup$
    – Alexander Etz
    Dec 6, 2015 at 22:57
  • $\begingroup$ If you post that as an answer I'll mark it as the right one! $\endgroup$
    – Pedro Carvalho
    Dec 7, 2015 at 1:48
  • $\begingroup$ There is no such thing as "the" noninformative prior... $\endgroup$
    – Xi'an
    Dec 7, 2015 at 7:22
  • $\begingroup$ I'm aware of that, that's why I said "a noninformative prior" then "noninformative priors" then "prior is noninformative." At no point did I express an uniqueness property of the priors. However, the theorem that states equivalence between confidence and credible intervals specifically mentions constant priors for location parametres and 1/s priors for scale parametres. $\endgroup$
    – Pedro Carvalho
    Dec 7, 2015 at 15:12

2 Answers 2

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You might look at this paper which has a few examples where they differ. One of the more interesting cases is with the effect size $\omega^2$, where the confidence interval often includes negative values (which are impossible values since it is a measure bounded from 0 to 1), but the credible interval never shows that pathology.

There is also a neat website built for the paper which lets you simulate the main results yourself.

Reference: Morey, R. D., Hoekstra, R., Rouder, J. N., Lee, M. D., & Wagenmakers, E.-J. (2015). The Fallacy of Placing Confidence in Confidence Intervals. Psychonomic Bulletin & Review, doi: 10.3758/s13423-015-0947-8.

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It all depends on the sample size. For large samples, the Bernstein von Mises theorem tells you that both approaches will be very close.

For small or moderate samples, the frequentist and Bayesian approaches may differ in some cases. This is usually referred to as the "Prior-Data conflict". This is a well-known and widely studied phenomenon. For some specific examples, please take a look at:

http://projecteuclid.org/download/pdf_1/euclid.ba/1340370946

This phenomenon is common on shape parameters with "highly informative" priors.

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  • $\begingroup$ Well yes, but my point was that I wanted something that was different even when the prior was noninformative. Having a normal prior for the mean of a normal likelihood is enough for the posterior credible interval to be different. $\endgroup$
    – Pedro Carvalho
    Dec 6, 2015 at 23:48

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