wmorgan comments on Take heed, for it is a trap - Less Wrong

47 Post author: Zed 14 August 2011 10:23AM

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Comment author: wmorgan 17 August 2011 06:02:01AM *  1 point [-]

We agree that we can't assign a probability to a property of a number without a prior distribution. And yet it seems like you're saying that it is nonetheless correct to assign a probability of truth to a statement without a prior distribution, and that the probability is 50% true, 50% false.

Doesn't the second statement follow from the first? Something like this:

  1. For any P, a nontrivial predicate on integers, and an integer n, Pr(P(n)) is undefined without a distribution on n.
  2. Define X(n), a predicate on integers, true if and only if the nth Godel number is true.
  3. Pr(X(n)) is undefined without a distribution on n.

Integers and statements are isomorphic. If you're saying that you can assign a probability to a statement without knowing anything about the statement, then you're saying that you can assign a probability to a property of a number without knowing anything about the number.

Comment author: komponisto 17 August 2011 06:34:43AM *  1 point [-]

We agree that we can't assign a probability to a property of a number without a prior distribution. And yet it seems like you're saying that it is nonetheless correct to assign a probability of truth to a statement without a prior distribution,

That is not what I claim. I take it for granted that all probability statements require a prior distribution. What I claim is that if the prior probability of a hypothesis evaluates to something other than 50%, then the prior distribution cannot be said to represent "total ignorance" of whether the hypothesis is true.

This is only important at the meta-level, where one is regarding the probability function as a variable -- such as in the context of modeling logical uncertainty, for example. It allows one to regard "calculating the prior probability" as a special case of "updating on evidence".

Comment author: wmorgan 17 August 2011 02:27:48PM 0 points [-]

I think I see what you're saying. You're saying that if you do the math out, Pr(S) comes out to 0.5, just like 0! = 1 or a^0 = 1, even though the situation is rare where you'd actually want to calculate those things (permutations of zero elements or the empty product, respectively). Do I understand you, at least?

I expect Pr(S) to come out to be undefined, but I'll work through it and see. Anyway, I'm not getting any karma for these comments, so I guess nobody wants to see them. I won't fill the channel with any more noise.

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