Does a $100 \times (1-\alpha)$% "exact" binomial CI imply use of the Clopper-Pearson method? I am working from a protocol to develop table specifications. The existing code which we have adapted for this purpose uses the Clopper-Pearson method to calculate the $100 \times (1-\alpha)$% confidence intervals. However, the protocol says that the "...exact $100 \times (1-\alpha)$% CI will be calculated." Is the Clopper-Pearson method exact? Or do they mean inverting the binomial likelihood of the response? If more than one method can be considered exact, in what sense is a method exact? Does it require that the lower and upper limits are observable proportions based on the sample size, e.g. with 10 observations allowable limits would be restricted to 0.0, 0.1, ..., 0.9, 1.0?
 A: The meaning of an "exact" confidence region is usually taken as an adjunct of the meaning of an "exact test" in hypothesis testing.  For a given confidence level $1-\alpha$ it requires that the required confidence level is met for every possible parameter value (without any approximating assumption being used):$^\dagger$
$$\mathbb{P}(\theta \in \text{CI}_\alpha(\mathbf{x}) | \theta) \geqslant 1-\alpha
\quad \quad \quad \quad \quad \text{for all } \theta \in \Theta.$$
Moreover, it would generally also require that there is no smaller mapping $\text{CI}_\alpha': \mathscr{X} \rightarrow \mathscr{P}(\Theta)$ that also meets this requirement (i.e., no mapping where $\text{CI}_\alpha'(\mathbf{x}) \subset \text{CI}_\alpha(\mathbf{x})$ for some $\mathbf{x} \in \mathscr{X}$).  In simple terms, this is saying that the "exact" confidence region meets the minimum confidence requirement for all possible parameter values (without any approximating assumption being made), and it cannot be made strictly smaller and still meet this requirement.
Any mapping $\text{CI}_\alpha: \mathscr{X} \rightarrow \mathscr{P}(\Theta)$ that meets the above requirement ---without using any approximating assumption--- could be considered an "exact confidence region" at confidence level $1-\alpha$.  If the mapping always yields a single connected interval then it is an "exact confidence interval".  The Clopper-Pearson method is a form of "exact" confidence interval with roughly equal tail sizes (usually).  Some people take this method to be synonymous with the notion of an "exact confidence interval", though strictly speaking it can be modified to use different tail area allocations and it will still be an "exact confidence interval".  It is not clear to me what you mean when you compare this method to "inverting the binomial likelihood of response" since this is roughly what the Clopper-Pearson interval already does.

$^\dagger$ I put the term "exact" in quotes throughout this discussion because it is actually quite conservative in many cases --- i.e., the probability of coverage of the true parameter is usually not exactly equal to $1-\alpha$ (but it is at least this large).
