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Misha comments on St. Petersburg Mugging Implies You Have Bounded Utility - Less Wrong
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Assigning zero probability to claims is bad because then one can't ever update to accept the claim no matter what evidence one has. Moreover, this doesn't seem to have much to do with "infinite claims" given that there are claims involving infinity that you would probably accept. For example, if we got what looked like a working Theory of Everything that implied that the universe is infinite, you'd probably assign a non-zero probability to the universe being infinite. You can't assign all hypotheses involving infinity zero probability if you want to be able to update to include them.
Suppose I randomly pick a coin from all of Coinspace and flip it. What probability do you assign to the coin landing heads? Probably around 1/2.
Now suppose I do the same thing, but pick N coins and flip them all. The probability that they all come up heads is roughly 1/2^N.
Suppose I halt time to allow this experiment to continue as long as we want, then keep flipping coins randomly picked from Coinspace until I get a tail. What is the probability I will never get a tail? It should be the limit of 1/2^N as N goes to infinity, which is 0. Events with probability of 0 are allowed -- indeed, expected -- when you are dealing with infinite probability spaces such as this one.
It's also not true that we can't ever update if our prior probability for something is 0. It is just that we need infinite evidence, which is a scary way of saying that the probability of receiving said evidence is also 0. For instance, if you flip coins infinitely many times, and I observe all but the first 10 and never see "tails" (which has a probability of 0 of happening) then my belief that all the coins landed "heads" has gone up from 0 to 1/2^10 = 1/1024.
There are only countably many hypotheses that one can consider. In the coin flip context as you've constructed the probability space there are uncountably many possible results. If one presumes that there's a really a Turing computable (or even just explicitly definable in some axiomatic framework like ZFC) set of possibilities for the behavior of the coin, then there are only countably many each with a finite probability. Obviously, this in some respects makes the math much ickier, so for most purposes it is more helpful to assume that the coin is really random.
Note also that your updating took an infinite amount of evidence (since you observed all but the first 10 flips) . So it is at least fair to say that if one assigns probability zero to something then one can't ever update in finite time, which is about as bad as not being able to update.
I introduced the concept of CoinSpace to make it clear that all the coinflips are independent of each other: if I were actually flipping a single coin I would assign it a nonzero (though very small) probability that it never lands "tails". Possibly I should have just said the independence assumption.
And yes, I agree that if we postulate a finite time condition, then P(X) = 0 means one can't ever update on X. However, in the context of this post, we don't have a finite time condition: God-TimFreeman explicitly needs to stop time in order to be able to flip the coin as many times as necessary. Once we have that, then we need to be able to assign probabilities of 0 to events that almost never happen.