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    $\begingroup$ Thanks. Just to be sure that I understand your answer correctly, do you mean that a simple solution, assuming we're in a frequentist framework, is to simply take into account the prior information when interpreting the CI?(rather than taking into account the prior info when computing the CI, which seems more complicated in a frequentist framework and may create various problems). $\endgroup$
    – Coris
    Commented Jul 3 at 15:18
  • $\begingroup$ Yes, that's what I'm saying. We already have an interpretation which seems satisfying to me, regardless of your prior bias. In a scientific context (and perhaps overall), I want to put a huge emphasis in making sure that my methods account and are explicit about bias, and avoid it if possible. Here, I don't see the value of adding bias, so I'd rather avoid it. $\endgroup$
    – Guillaume Dehaene
    Commented Jul 3 at 15:35
  • $\begingroup$ NB: biasing a CI sounds impossible, though I'm sure some unnatural construction exists. However, it is straightforward to have biased intervals in a frequentist framework. We just need to consider a formulation for interval estimation based on some sort of score function relating the interval to the true underlying $p$. It is then straightforward to consider biased estimators which put more more importance in having low score in some regions. $\endgroup$
    – Guillaume Dehaene
    Commented Jul 3 at 15:39