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Cyan comments on A Request for Open Problems - Less Wrong
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That's it. The so-called 50% confidence interval sometimes contains c with certainty. Also, when x_max - x_min is much smaller than 0.5, 50% is a lousy summary of the confidence (ETA: common usage confidence, not frequentist confidence) that c lies between them.
If it's less than 0.5, is the confidence simply that value times 2?
"Confidence" in the statistics sense doesn't always have much to do with how confident you are in the conclusion. Something that's the real line in half of all cases and the empty set in the other half of all cases is a 50% confidence interval, but that doesn't mean you're ever 50% confident (in the colloquial sense) that the parameter is on the real line or that the parameter is in the empty set.
The Credible interval article on Wikipedia describes the distinction between frequentist and Bayesian confidence intervals.
The general pattern here is that there's something you do care about and something you don't care about, and frequentism doesn't allow you to talk about the thing you do care about, so it renames the thing you don't care about in such a way as to suggest that it's the thing you do care about, and everyone who doesn't understand statistics well interprets it as such.
The interesting thing about the confidence interval I'm writing about is that it has some frequentist optimality properties. ("Uniformly most accurate", if anyone cares.)
Well. So if all men were within 10 cm of each other, and uniformly distributed, and we plucked 2 random men out, and they were 4cm apart, would c be between them with 80% probability? Or some other value?
The shorter man can be between c-5 and c+1 with all values equally probable, if he's between c-5 and c-4 or c and c+1 then c is not between them, if he's between c-4 and c then c is between them, so assuming a uniform prior for c the probability is 2/3 if I'm not mistaken.
Ah, I see what I did wrong. I think.
Yup. Under the uniform prior the posterior probability that c is between the two values is d/(1 - d), 0 < d < 0.5, where d = x_max - x_min (and the width of the uniform data distribution is 1).
The answer to that depends on what you know about c beforehand -- your prior probability for c.
Whoops -- "confidence" is frequentist jargon. I'll just say that any better method ought to take x_max - x_min into account.