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

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

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Comment author: sharpneli 14 December 2009 07:37:56AM 0 points [-]

As there is the 1:1 mapping between set of all reals and unit interval we can just use the unit interval and define a uniform mapping there. As whatever distribution you choose we can map it into unit interval as Pengvado said.

In case of set of all integers I'm not completely certain. But I'd look at the set of computable reals which we can use for much of mathematics. Normal calculus can be done with just computable reals (set of all numbers where there is an algorithm which provides arbitrary decimal in a finite time). So basically we have a mapping from computable reals on unit interval into set of all integers.

Another question is that is the uniform distribution the entropy maximising distribution when we consider set of all integers?

From a physical standpoint why are you interested in countably infinite probability distributions? If we assume discrete physical laws we'd have finite amount of possible worlds, on the other hand if we assume continuous we'd have uncountably infinite amount which can be mapped into unit interval.

From the top of my head I can imagine set of discrete worlds of all sizes which would be countably infinite. What other kinds of worlds there could be where this would be relevant?