# When do (and don't) confidence intervals and credible intervals coincide?

Yes, I know there are many questions on comparing these two types of intervals, but none of them appear to answer this exact question.

Here is a blog post demonstrating one case where the two intervals are the same and one where they are different:

https://jakevdp.github.io/blog/2014/06/12/frequentism-and-bayesianism-3-confidence-credibility/

But are there any general rules to this? My personal experience suggests that misinterpreting confidence intervals as credible intervals is widespread (I would guess that near 100% are misinterpreted in the medical literature). However, there seems to be no problem with this in practice as long as the confidence interval is a close approximation to the credible interval.

• Could you expand on how the answer to Examples of when confidence interval and credible interval coincide doesn't answer this exact question? – Nick Stauner Aug 29 '14 at 3:35
• @NickStauner That is only one example. In a comment probabilityislogic hints at further situations. I would like to know when to be concerned about this misinterpretation. For example, I suspect it is uncommon for unimodal distributions to accurately describe biological systems. What about in that case? I am wondering if there is a general rule though. – Livid Aug 29 '14 at 3:45

I'm not sure you can consider this a complete answer so you can double check yourself, however, here goes.

By the definition of confidence intervals that

there's an $X\%$ chance that when computing the $X\%$ confidence intervals (CI) the true value $y$ will fall within computed CI,

then you can synthesize an experiment where you know the true parameter values $y$, and you simulate the noise (based on the assumed likelihood function) let's say $P=1000$ times. When you do the fit and compute the $X\%$ confidence intervals, $y$ should fall within the CIs $X\%$ of the time. If this fails or succeeds in a significant way, then it affects a decision.

On the other hand, given the definition of the credible intervals where

Given the observed data, there is a $X\%$ probability that the true value $y$ falls within the $X\%$ credible interval

it means that

• you must synthesize $P$ different parameters $\{y_p\}_{p=1,\ldots,P}$ (which are your true values),
• solve using a Bayesian estimator $P$ times,
• compute the $X\%$ credible intervals for each ($P$ times),
• and expect that $X\%$ of the true values $y_p$, should fall within the credible intervals.

Note: $P$ and $X$ are the same in the aforementioned scenarios.

So to summarize, to be able to compare credible intervals to confidence intervals fairly, you need to follow their definitions. In the frequentist approach you assume a fixed set of parameters (remember frequentists assume parameters are fixed) and simulate noise in the measurements (data), whereas in the Bayesian approach you assume your data is fixed, so you must randomize'' the parameters. If you follow this approach credible and confidence intervals can be compared fairly (no matter the prior distribution).