Timeline for What is a generalized confidence interval?
Current License: CC BY-SA 4.0
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| Nov 30, 2020 at 18:00 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Nov 27, 2020 at 13:07 | comment | added | Sextus Empiricus | I agree with you that the chosen prior is not always 'correct'. So considering a credible interval as always a generalized confidence interval is not correct either. But consider it just as an example about the concept of a generalized confidence interval. A generalized confidence interval contains the parameter 95% of the time, whereas a confidence interval contains the parameter 95% of the time conditional on the true parameter. | |
| Nov 27, 2020 at 13:07 | comment | added | Sextus Empiricus | @StéphaneLaurent consider a problem statement like this $$X \sim N(\theta,1) \quad \text{where} \quad \theta \sim N(0,\tau^2)$$ which is used in the question and answer here. In this case the correct prior for $\theta$ is a normal distribution with mean $0$ variance $\tau^2$ (and if we know the problem well enough then this might be even a practical situation and not merely a theoretical one). | |
| Nov 27, 2020 at 13:00 | comment | added | Stéphane Laurent | And what is a correct prior? And do you have some references to support your claims? | |
| Nov 27, 2020 at 12:59 | comment | added | Sextus Empiricus | @StéphaneLaurent of course the model (like the prior) needs to be correct. The same is true for confidence intervals, they will also be wrong if the model (e.g. the likelihood function) is incorrect. | |
| Nov 27, 2020 at 12:56 | comment | added | Stéphane Laurent | That is not true. An obvious counter-example is when you take a prior distribution whose support does not contain the true value of the parameter. Then the posterior interval never catches the true value. | |
| Nov 27, 2020 at 12:51 | comment | added | Sextus Empiricus | @StéphaneLaurent I would disagree that it has not anything to do with it. I believe that a Bayesian credible interval is an example of a generalized confidence interval, because it satisfies the condition from the theorem (if the prior is correct). A credible interval will not contain the parameter 95% of the time for a specific $\theta_k$ (So it is not a confidence interval). But for all $\theta_k$ it will on average be correct 95% of the time. | |
| Nov 27, 2020 at 12:39 | comment | added | Stéphane Laurent | Sorry I don't get your point. Why are you introducing the Bayesian stuff? This has nothing to do with the question. | |
| Nov 27, 2020 at 12:33 | history | answered | Sextus Empiricus | CC BY-SA 4.0 |