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Cyan comments on A Request for Open Problems - Less Wrong
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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.