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Cyan comments on Against improper priors - Less Wrong Discussion
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Suppose I am sampling from a normal distribution. Is it legitimate to declare that my prior information is such that no matter what values I observe for the first two data points, after observing them my posterior predictive probability for the event "the third data point lies between the first two data points" is 50%?
It's coherent to act that way, but it's improper to call it a prior. If you had a proper prior, you'd bet on any question at any time, not just until after seeing two points. But you knew that, right?
Well, the condition I describe in words above doesn't directly limit my ability to bet on any question at any time -- it just specifies one possible bet in one particular set of states of information.
However, the condition can be turned into an integral equation in which the prior density is the unknown quantity. The equation can be explicitly solved to give an analytical expression for the unique prior density which satisfies the verbal description above. Since I posted the question in this thread, you can probably guess the punch-line: the prior is improper. In fact, it's the standard "non-informative" prior for the normal distribution with unknown mean and variance.