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rebellionkid comments on Einstein's Arrogance - Less Wrong
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There are not an infinite number of possible hypotheses in a great many sensible situations. For example, suppose the question is "who murdered Fred?", because we have already learned that he was murdered. The already known answer: "A human alive at the time he died.", makes the set finite. If we can determine when and where he died, the number of suspects can typically be reduced to dozens or hundreds. Limiting to someone capable of carrying out the means of death may cut 90% of them.
To the extent that "bits" of evidence means things that we don't know yet, the number of bits can be much smaller than suggested. To the extent that "bits" of evidence includes everything we know so far, we all have trillions of bits already in our brains and the minimal number is meaningless.
What about the aliens who landed on earth, murdered Fred and then went away again? Or the infinite number of other possibilities, each of which has a very small probability?
What confuses me about this is that, if we do accept that there are an infinite number of possibilities, most of the possibilities must have an infinitesimal probability in order for everything to sum to 1. And I don't really understand the concept of an infinitesimal probability -- after all, even my example above must have some finite probability attached?
Just to point out what may be a nitpick or a clarification. It's perfectly possible for infinity many positive things to sum to a finite number. 1/2+1/4+1/8+...=1.
There can be infinitely many potential murderers. But if the probability of each having done it drops off fast enough you can avoid anything that is literally infinitesimal. Almost all will be less than 1/3^^^^^^3 of course, but that's a perfectly well defined number you know how to do maths with.
Hate to nitpick myself, but 1/2+1/4+1/8+... diverges (e.g., by the harmonic series test). Sum 1/n^2 = 1/4 + 1/9 + ... = (pi^2)/6 is a more fitting example.
An interesting question, in this context, is what it would mean for infinitely many possibilities to exist in a "finite space about any point that can be reached at sub-speed of light times." Would it be possible under the assumption of a discrete universe (a universe decomposable no further than the smallest, indivisible pieces)? This is an issue we don't have to worry about in dealing with the infinite sums of numbers that converge to a finite number.
That's not correct at all. sum(1/2^n)[1:infinity] = 1.
Oops, misread that as sum(1/(2n))[1:infinity] (which it wasn't), my bad.