After stumbling upon the concept in a statistics textbook, I tried to wrap my head about it, and finally came to a conclusion which seems to fit all the explanations which I have seen so far: A credible interval is what non-statisticians think a confidence interval is.
Digression for those like me-from-an-hour-ago who don’t know the difference
If we observed data and predicted some parameter from it, let’s say the mean μ, the credible interval is the interval [μmin, μmax] for which we are 95% sure that mu falls inside (or some number other than 95%, if we used another level). The confidence interval taught in introductory statistics classes can overlap with the credible interval, but will not always overlap well. If you want to brave the explanation, try reading this and this question on Cross Validated; what helped me finally understand, after much head-scratching, was this answer.
Does it mean that it would be scientifically preferable to use a credible interval over a confidence interval in my results? If yes, why haven’t I seen any publications which use it?
- Is it because the concept should be used, but the measuring scientists have not yet caught up with the correct statistic methods?
- Or is the meaning of the original confidence interval better suited to explaining results from empirical studies?
- Or is it that in practice, they are so often overlapping that it doesn’t matter at all?
- Does the choice depend on the statistical distribution we are assuming for our data? Maybe with a Gaussian distribution, they always overlap numerically, so nobody outside of pure statistic cares about the difference (many studies I have read don’t even bother to calculate any kind of interval, and maybe around 1% ever give space to the thought that their data might not be normally distributed).
- Does it depend on our scientific-theory standing? For example, it feels like the confidence interval should be used in positivist work and the credible interval in interpretivist work, but I am not sure that this feeling is correct.
The type of interval indicates what type of method you used. If a credible interval (or Bayesian variant), it means a Bayesian method was used. If a confidence interval, then a frequentist method was used.
Re: Or is it that in practice, they are so often overlapping that it doesn’t matter at all?
As long as
- the conditions to use methods are reasonably satisfied (e.g. “independence of observations” is a requirement for many methods),
- the Bayesian method doesn’t use an informative prior,
- the sample that isn’t very small, and
- the models / methods are analogous,
the credible and confidence intervals will be close to each other. The reason: the likelihood will dominate the Bayesian prior, and the likelihood is what is typically used in frequentist methods.
I would suggest not fretting about which to use. If you want an informative prior, then be sure to use a Bayesian method. If not, then choose a suitable method and context (frequentist or Bayesian), check to make sure the conditions required to apply the method are reasonably satisfied (so important but so rarely done!), and then move forward if the method is appropriate for the type of data.