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In Computer Age Statistical Inference (open source access in this link), on page 5 (pasted below for reference), it shows the following highlighted part about confidence intervals. Based on a couple posts on this site, such as What's the difference between a confidence interval and a credible interval?, this book's description appears to be wrong?

The author claims the confidence interval calculated here has a 95% chance of containing the true value. Doesn't 95% confidence actually mean that 95% of confidence intervals calculated from these random samples will contain the true value?

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  • $\begingroup$ Yes, the author's interpretation of the confidence interval is technically incorrect. $\endgroup$ – JTH Jul 8 at 18:31
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    $\begingroup$ If the authors [plural in this case] are Efron and Hastie I would always plump for the interpretation that they are being informal rather than wrong. $\endgroup$ – Nick Cox Aug 7 at 10:39
  • $\begingroup$ @NickCox Yup those are the authors. I only read a couple of pages from this particular book, but Hastie also was one of the authors on ESL, and I find some informality there as well -- In fact, I really struggled with some of their terminology and definitions for all sorts of different errors in this particular chapter (can't remember which one it was) and that was really confusing for me. I don't have a formal stats background, so this kind of informality is not good for me. $\endgroup$ – anonuser01 Aug 7 at 20:06
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A confidence interval estimate relates to an interval that has $\alpha \%$ probability to contain the parameter, conditional on the parameter.

This contrasts with a credible interval, which has $\alpha \%$ probability to contain the parameter, conditional on the observation.

See the example from this question/answer where a comparison is made between credible intervals and confidence intervals for estimating a parameter $\theta$ based on observation $X$. For a given prior and probability density of the observations we can plot the frequency of the success of the interval as function of the observation and as function of the true parameter.

Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

So you see on the left image that a 95% Confidence Interval does (in some sense) imply a 95% chance of containing the mean.

But, that is when you condition on the value $\theta$. Conditional on a given observation $X$ (the right side plot) it might not be true, and the confidence interval will have different chances for different observations.

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