The absence of comments here doesn’t reflect well on us, but this is a tricky topic. I’m honestly trying to get to the bottom of this and the bottom ain’t in sight yet.
EDIT: I’m not sure a prior that matched confidence intervals would be a good thing. See point III.b “Truncated exponential distribution” in this pdf for an example where a 90% confidence interval gives a result that’s actually logically ruled out by the sample. (Cyan, am I restating obvious stuff? Too stupid to say for sure yet.)
To be honest, I’m not shocked that most people aren’t equipped to or interested in grappling with this stuff. If I weren’t a Bayesian working for a frequentist I wouldn’t be thinking so much about why frequentists do what they do. I was hoping that the more mathematically inclined folks would find this argument startling enough to try to knock it down—I’d be happy to be wrong.
It isn’t so much that we want posterior intervals to match some crappy-arsed confidence interval. We just want them to be calibrated, and as near as I can tell, calibration is equivalent to valid confidence coverage. We know from results in the literature on matching priors that posterior intervals aren’t calibrated in general (provided I’ve got the equivalence of calibration and valid confidence coverage right) . So we can have calibration, or we can have rational decisions, but not both (?).
I suspect it’s an issue of jargon or technical difficulty. Like I mentioned, my math background is at least decent, and I have some serious trouble wrapping my mind around what issue is being debated here. I strongly suspect that has more to do with how the issue is presented and explained than with the issue itself, though I could be quite wrong.
There’s got to be a way to express this in plain English (or even plain math); how, for example, do a frequentist and a Bayesian see the same problem differently, and why should we care?
If you pick out some of the specific jargon words that are opaque to you, I can taboo them or provide links and we’ll can see if I can revise the posts into comprehensibility.
The absence of comments here doesn’t reflect well on us, but this is a tricky topic. I’m honestly trying to get to the bottom of this and the bottom ain’t in sight yet.
EDIT: I’m not sure a prior that matched confidence intervals would be a good thing. See point III.b “Truncated exponential distribution” in this pdf for an example where a 90% confidence interval gives a result that’s actually logically ruled out by the sample. (Cyan, am I restating obvious stuff? Too stupid to say for sure yet.)
To be honest, I’m not shocked that most people aren’t equipped to or interested in grappling with this stuff. If I weren’t a Bayesian working for a frequentist I wouldn’t be thinking so much about why frequentists do what they do. I was hoping that the more mathematically inclined folks would find this argument startling enough to try to knock it down—I’d be happy to be wrong.
It isn’t so much that we want posterior intervals to match some crappy-arsed confidence interval. We just want them to be calibrated, and as near as I can tell, calibration is equivalent to valid confidence coverage. We know from results in the literature on matching priors that posterior intervals aren’t calibrated in general (provided I’ve got the equivalence of calibration and valid confidence coverage right) . So we can have calibration, or we can have rational decisions, but not both (?).
I suspect it’s an issue of jargon or technical difficulty. Like I mentioned, my math background is at least decent, and I have some serious trouble wrapping my mind around what issue is being debated here. I strongly suspect that has more to do with how the issue is presented and explained than with the issue itself, though I could be quite wrong.
There’s got to be a way to express this in plain English (or even plain math); how, for example, do a frequentist and a Bayesian see the same problem differently, and why should we care?
If you pick out some of the specific jargon words that are opaque to you, I can taboo them or provide links and we’ll can see if I can revise the posts into comprehensibility.