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pengvado comments on What Are Probabilities, Anyway? - Less Wrong

24 Post author: Wei_Dai 11 December 2009 12:25AM

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Comment author: pengvado 14 December 2009 04:38:02AM 2 points [-]

There is a 1:1 mapping between "the set of reals in [0,1]" and "the set of all reals". So take your uniform distribution on [0,1] and put it through such a mapping... and the result is non-uniform. Which pretty much kills the idea of "uniform <=> each element has the same probability as each other".

There is no such thing as a continuous distribution on a set alone, it has to be on a metric space. Even if you make a metric space out of the set of all possible universes, that doesn't give you a universal prior, because you have to choose what metric it should be uniform with respect to.

(Can you have a uniform "continuous" distribution without a continuum? The rationals in [0,1]?)