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    $\begingroup$ @whuber thanks for the feedback, I have removed the aside. I do see that it answers a different question. $\endgroup$ Commented Jul 12, 2024 at 14:52
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    $\begingroup$ Similarly, in formulas if X is an rv then x is used to denote a particular sample. So in the CI context, $\hat{\theta}_{L} $ is a notation abuse, it would be clearer to write $\hat{\theta}_{L}(X)$ or $\hat{\theta}_{L}(x)$ to diffentiate between the procedure and its outcome. $\endgroup$ Commented Jul 12, 2024 at 18:47
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    $\begingroup$ And since the answer is in that similar vein, I am leaving a +1 (who downvoted it anyway, hello?). $\endgroup$ Commented Jul 13, 2024 at 3:52
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    $\begingroup$ @GrahamBornholt the fact that I accidentally misinterpreted you makes my point here :) $\endgroup$ Commented Jul 13, 2024 at 7:26
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    $\begingroup$ Great answer! MacKay's book <inference.org.uk/mackay/itila/book.html> in §37.3 gives several simple examples illustrating these points. $\endgroup$ Commented Sep 23, 2024 at 20:54