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I understand that pre-data collection we can be 95% confident that the interval we're about to calculate will contain the true population parameter θ.

This doesn't mean that once we've calculated the C.I. that there's a 95% chance that θ is in the C.I.
θ is either contained within the interval or it isn't. As now both θ and the C.I. are fixed quantities

However, if we wished to calculate an interval with a 95% probability that that interval contained θ we could use a Bayesian credible interval.

I've seen this discussion in many places. These discussions also tend to include how often the misinterpretation of C.I. appear in the research literature and how few researches understand the correct interpretation when polled, or quizzed on the topic.

What I've failed to find is the real-world consequences of interpreting a C.I. as if it were a credible interval.

Can someone please point me in the direction of real-world consequences? Preferably a case-study, but even an example would be greatly appreciated.

Thank You.

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  • $\begingroup$ In Bayesian, $\theta$ is a random variable. After the interval is established, it is fixed as you said. Then the random $\theta$ can jump in to the interval today, and jump out of the interval tomorrow. I got this idea from a paper published on Significance. $\endgroup$ – user158565 Jul 31 at 2:46
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Where the confidence and credible intervals have similar bounds it is hard to see that interpreting one as if it were the other is an important problem.

I have not come across a real-world circumstance where the distinction between a fixed unknown parameter value and a parameter value that is a random variable is problematical of itself. However, it is important to realise that credible intervals can be shaped by an informative prior and such an interval will be different from the frequentist confidence interval derived for the same data. In that circumstance it would be more problematical to assume that the confidence interval reflected the (posterior) probability of the parameter values of interest.

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