Timeline for Are there any examples where Bayesian credible intervals are obviously inferior to frequentist confidence intervals
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| Jun 11, 2020 at 14:32 | history | edited | CommunityBot |
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| Jan 11, 2020 at 8:45 | comment | added | Sextus Empiricus | @probabiltyislogic I agree with you that one can mock the "a 95% Confidence Interval (CI) does not imply a 95% chance of containing the mean" as Jaynes does in the article. It is often not the probability that is interresting (unless one does the test many times on a large ensemble such that focussing on frequency of success makes sense, e.g. quality testing or evaluating stocks, or when the loss function depends on the true $\theta$ and not on the observed $X$). However the creation of a statement about posterior probability is not a real solution when the prior is not correct. | |
| Jan 11, 2020 at 8:34 | comment | added | Sextus Empiricus | @probabiltyislogic you are right the likelihood is not always the least problematic. I should have stated that sometimes $p (\theta) $ is the biggest problem, sometimes it is the $p (X\,\vert\,\theta) $ sometimes it is both. But besides that, it's probably not what makes people choose, right or wrong, for the frequentist method (the essential difference is in how they draw interval boundaries and choose to make the probability that the interval is correct dependent on other parameters; as illustrated in the two graphs that I made based on the figure from Wasserman ). | |
| Jan 11, 2020 at 3:43 | comment | added | probabilityislogic | I disagree here - often the likelihood is where the real problems are (e.g. constant variance assumption). Why is there a huge literature on "outliers" and "robustness" if likelihoods are reasonable? Additionally, the 'problem' with the prior can be easily fixed, by using a t-distribution with low df instead of normal. For large "true values" of $\theta$ the prior would be ignored with the posterior concentrating around $X$ rather than $cX$. | |
| Jan 10, 2020 at 11:50 | comment | added | Sextus Empiricus | @probabilityislogic When one constructs a confidence interval then one uses the model $p(X \, \vert \, \theta)$. When one constructs a credible interval then one has also an additional model/assumption/believe for the marginal distribution $p(\theta)$. Indeed, for both assumptions/models we should be considering how trustworthy they are and by how much the errors in the assumptions propagate into the idealistic expressions of Bayesian/frequentist probability. Luckily the expression for $p(X \, \vert \, \theta)$ is often very reasonable, but the $p(\theta)$ is not always so clear. | |
| Jan 10, 2020 at 11:35 | comment | added | probabilityislogic | just on your comment on the incorrect prior assumptions - if we relax this, then we also should be considering that the model $p(X|\theta)$ is also "wrong". But this usually is not helpful to anyone - the solution is usually an implicit version of "change the model" (e.g. non-parametric tests, etc) | |
| Jan 9, 2020 at 17:19 | comment | added | Sextus Empiricus | ....So if you use the same variability you can always make the intervals shorter or at least the same size. But when you make a constant α% dependency on X, as with a typical credible interval, then it might be possible that the credible interval is not smaller than the confidence interval for every X. That means that the credible interval does not always dominate the confidence interval. (I have no clear example in mind, but I imagine it must be possible) | |
| Jan 9, 2020 at 17:17 | comment | added | Sextus Empiricus | When I write "So it may not always be optimally selecting the shortest interval, and for each observation $X$ it may be possible to decrease the length of the interval by shifting the boundaries while enclosing the same α% amount of probability mass." It must be noted that this α% is variable as function of X for the confidence interval... | |
| Jan 9, 2020 at 16:16 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Jan 9, 2020 at 16:04 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Jan 9, 2020 at 15:28 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Jan 9, 2020 at 15:15 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Jan 9, 2020 at 15:08 | history | answered | Sextus Empiricus | CC BY-SA 4.0 |