Timeline for Are there any examples where Bayesian credible intervals are obviously inferior to frequentist confidence intervals
Current License: CC BY-SA 4.0
11 events
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| S Jul 3, 2022 at 15:40 | history | suggested | Glorfindel | CC BY-SA 4.0 | broken link fixed, cf. https://math.meta.stackexchange.com/a/34713/228959 |
| Jul 3, 2022 at 9:09 | review | Suggested edits | |||
| S Jul 3, 2022 at 15:40 | |||||
| Oct 2, 2012 at 8:51 | comment | added | probabilityislogic | The problem you speak of is more about prior specification than an error. We want the prior to accurately describe what information you have. The above example is one where we consider the likelihood function is more reliable than the prior. | |
| Oct 2, 2012 at 8:43 | comment | added | probabilityislogic | Note that the prior being off by an order of magnitude doesn't matter so long as the tails of the prior are "fatter" than the tails in the likelihood. For example, if you have $p(x_i|\mu)\sim N(\mu,1)$ for $i=1,\dots,n$ and you set your prior as $p(\mu)\sim Cauchy(m,v)$. Then the posterior mean cannot be more than some fixed distance away from the sample mean. Further the distance tends to zero as $|m-\overline{x}|\to\infty$ - ie as our prior guess becomes more in conflict with the data. | |
| Apr 10, 2012 at 17:47 | comment | added | Stéphane Laurent | Maybe I should edit my answer and delete my "example" - this is not the serious part of my answer. My answer mainly was about the meaning of "the" Bayesian approach. What do you call the Bayesian approach ? This approach requires the choice of a subjective prior or it uses an automatic way to select a noninformative prior ? In the second case it is important to mention the work of Bernardo. Secondly you have not defined the "superiority" relation between intervals: when do you say an interval is superior to another one ? | |
| Apr 10, 2012 at 10:36 | comment | added | Dikran Marsupial | @cardinal it isn't evading criticism of Bayesian methods, of course choice of prior is an issue. It just isn't the issue that is relevant to this particular question. The difficulty of performing the integrals is another weakness of Bayesian methods. Horses for courses, the trick is to know which horse for which course, hence my interest in the question. | |
| Apr 10, 2012 at 10:33 | comment | added | Dikran Marsupial | Cardinal and guest are correct, my question explicitly included "Examples based on incorrect prior assumptions are not acceptable as they say nothing about the internal consistency of the different approaches." for a good reason. Frequentist tests can be based on incorrect assumptions as well as Bayesian ones (the Bayesian framework states the assumptions more explicitly); the question is whether the framework has weaknesses. Also if the true value was in the prior, but not the posterior, that would imply that the observations ruled out the possibility of the true value being correct! | |
| Apr 7, 2012 at 13:14 | comment | added | Stéphane Laurent | My "example" was not the important part of my answer. But what is a good choice of prior ? It is easy to imagine a prior whose support contains the true parameter but the posterior does not, so the frequentist interval is superior ? | |
| Apr 6, 2012 at 20:14 | comment | added | guest | @cardinal's comment is quite right. The prior here is off by an order of magnitude, making the criticism very weak. Prior information matters to frequentists too; what one knows a priori should determine e.g. what estimates and test statistics are used. If these choices are based on information that's wrong by an order of magnitude, poor results should be expected; being Bayesian or frequentist doesn't come into it. | |
| Apr 6, 2012 at 19:42 | comment | added | cardinal | I am speculating, but I suspect this answer is bound to get the same treatment that others have. Someone will simply argue this is an issue of poor choice of prior and not of some inherent weakness of Bayesian procedures, which in my view partially tries to evade a valid criticism. | |
| Apr 6, 2012 at 19:30 | history | answered | Stéphane Laurent | CC BY-SA 3.0 |