I just read the following sentence from Wikipedia:

A 95% confidence interval does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).

Instead, the correct interpretation would be

The confidence interval can be expressed in terms of samples (or repeated samples): "Were this procedure to be repeated on numerous samples, the fraction of calculated confidence intervals (which would differ for each sample) that encompass the true population parameter would tend toward 90%.

But then my question is: if the current confidence intervals are an instance of an procedure that contains the true parameter value 90% of the time when time tends to infinity, like the second quote implies, then why can't we say that the current intervals contains the true parameter with 90% probability (the probability referring to the CI calculation, not to the parameter)? Isn't that the frequentest definition of probability?

  • $\begingroup$ The current interval either contains or does not contain the true population parameter. The probability is either 0 or 1, you just don't know which it is. $\endgroup$ – Heteroskedastic Jim Oct 25 '18 at 23:56
  • $\begingroup$ Ok, I get that, but since the intervals depend on the sample, we can talk in probabilistic terms about them, can't we? In other words, where is the error in my question from the last paragraph? $\endgroup$ – nestor556 Oct 26 '18 at 0:00
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    $\begingroup$ You can if you want to but that's not what the probability attached to the width of the confidence interval means. The probability attached to the width of the CI is only true in that sense before the study. $\endgroup$ – Heteroskedastic Jim Oct 26 '18 at 0:04
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    $\begingroup$ If you want to make precise claims about the probability that the interval includes the parameter, then you need to calculate the interval under different assumptions. See stats.stackexchange.com/questions/2272/… $\endgroup$ – Heteroskedastic Jim Oct 26 '18 at 0:09
  • $\begingroup$ You may also like to look up references to prediction intervals which take account of the standard error of the mean for the sample used to construct the confidence interval $\endgroup$ – Robert Jones Oct 26 '18 at 21:25

Suppose I build a confidence interval $(L, U)$ that covers the circle constant $\pi = 3.14159265...$ with probability 95%. I get the result $(3.2, 3.3)$. Can you see the issue with saying that $\pi$ is between 3.2 and 3.3 with 95% probability? Clearly it is in the interval with 0% probability, not 95%. The wrong definition leads to a nonsense answer in this case. This is a defining feature of Frequentist inference: the target parameter is fixed. In practice it is usually unknown as well, but in this case it is known to be exactly $\pi$.

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    $\begingroup$ I'm not sure I understand this example, because you could make the exact same claim about a 95% credibility interval $\endgroup$ – Frans Rodenburg Oct 26 '18 at 1:56
  • $\begingroup$ @Frans Bayesian inference is a rational system of reasoning. Consequently, you cannot place priors on fixed mathematical constants like $\pi$. Or, rather, the only mathematically consistent prior you can put on $\pi$ is a point mass at $\pi$. $\endgroup$ – guy Oct 26 '18 at 2:01
  • $\begingroup$ @Frans see also the answer by Xi'an here, and the associated question, which makes clear the issue of putting a prior on a known mathematical constant. The point of my answer is that one can build a confidence interval for the fixed constant $\pi$ which satisfies all the definitions of a confidence interval. You cannot construct a prior for a fixed constant, and trying to do so leads to contradiction. $\endgroup$ – guy Oct 26 '18 at 15:40

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