# Why is a Frequentist confidence interval usually referred to as “exact” in comparison to the Bayesian posterior probability interval?

I have noticed that a lot of literature usually refers to the frequentist confidence interval as "exact" in comparison to the Bayesian probability/credibility interval, which is calculated off the posterior distribution. In this sense, are Bayesian probability intervals viewed as an approximation of the frequentist interval? This doesn't make intuitive sense as the Bayesian and Frequentist intervals seem to be inherently different. Making things more confusing, I know that sometimes the Bayesian and Frequentist intervals will match up:

For example, for a case where the 95% probability interval is also a 95% "exact" confidence interval is the case where we assume $X_1, \ldots, X_n \sim \mathcal{N}(\mu, \sigma_1^2)$ are iid, and that $Y_1, \ldots, Y_n \sim \mathcal{N}(\mu, \sigma_2^2)$, the marginal posterior distribution for $\mu$ has a 95% probability interval (bayesian) that is also the same as the "exact" 95% confidence interval (frequentist).

Would anyone be able to help me see where my understanding is wrong or confused? Thanks!

• Doesn't "exact" in this case refer to confidence intervals built off of exact tests (en.wikipedia.org/wiki/Exact_test)? Given that Bayesian inference is usually not concerned with rejecting null hypotheses, such terminology wouldn't be applicable. – C.R. Peterson Dec 9 '15 at 11:27
• I would not put too much value in this denomination of "exact": a 96% confidence interval is exact in the sense that it contains the true value of the parameter for 96% of the samples. While the Bayesian 96% credible interval contains the parameter with probability 0.96 under the posterior, for a given realisation of the sample. They match up in some rare cases and almost always asymptotically in the sample size $n$. – Xi'an Dec 9 '15 at 20:47