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Matt_Simpson comments on Probability puzzle - Less Wrong Discussion
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Well, you could use an improper prior. The measure exists, it just isn't a probability measure. The prior is over the ratio of heads to tails, which is a postitive, unbounded real valued number, so U(0,1) is certainly not appropriate. However, this is certainly not the prior you would use for the correct Bayesian calculation, though it may be useful as an approximation.