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

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

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Comment author: Wei_Dai 13 December 2009 10:56:07PM 4 points [-]

My understanding is that it's possible to have a uniform distribution over a finite set, or an interval of the reals, but not over all integers, or all reals, which is why I said in the sentence before the one you quotes, "suppose there is one possible world for each integer in the set of all integers."

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]?)

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?