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I'm learning about confidence intervals and it explicitly states that the confidence interval is not to be interpreted as that it has 95% probability of containing the true value of the parameter.

However if I understand the definition correctly it says that if you have infinite samples then the 95% proportion of those samples contain the true parameter.

Why can't this proportion be interpreted as a probability? E.g. if you have 60 black sheep and 40 white sheep, you have 60% chance of picking a black sheep etcetera

Richard Hardy
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bananenheld
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    "Why can't this proportion be interpreted as a probability?" it can, but to do so you are switching from a frequentist definition of a probability (a long run frequency) to a Bayesian one (degree of plausibility). You can't assign a frequentist probability to a particular event (the true value is in this interval because that has no long run frequency, it is either in the interval or it isn't). – Dikran Marsupial May 21 '22 at 08:40
  • @Dikran Marsupial Thank you very much!! That is such a clear explanation. How can I make it an official answer because the answers in the link of COOLSerdash are so convoluted and unintuitive. Maybe that helps people in the future in finding a good answer to this question? – bananenheld May 21 '22 at 09:19

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As an illustration, consider the following method of producing a confidence interval:

Based on some arbitrary random variable (e.g. rolling a dice) we output the interval $(-\infty,+\infty)$ with probability 95%. Otherwise, we output the interval that contain the number $\pi$ only. This is an exact 95% confidence interval - for whatever parameter - because it will contain the true value exactly 95% of the time.

Obviously however, when the interval is $(-\infty,+\infty)$ we know that it contains the true value with absolute certainty, and when it is $[\pi]$ we know with (almost) absolute certainty that it doesn't.

J. Delaney
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