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Manfred comments on Questions of Reasoning under Logical Uncertainty - Less Wrong
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Comments (19)
Your description of "true arithmetic" seems to be unusually realist. Do you mean that, or is it meant to illustrate a way of thinking about what a logical probability on models is a 'probability of'?
Pg. 5, 2^|−ψ| should be 2^-|ψ|.
I'm not sold on your discussion of desiderata for priors. Specifically, on the properties that a prior can have without its computable approximation having them - I almost feel like these should be split off into a separate section on desiderata for uncomputable priors over models. Because the probability distribution over models should update on deduction as the agent expends computational resources, I don't think one can usefully talk about the properties of a (actually computed) prior alone in the limit of large resources. The entire algorithm for assigning logical probability is, I think, the thing whose limit should be checked for the desiderata for uncomputable priors.
Yes, the description of true arithmetic is a bit realist; I'm not particularly sold on the realism of true arithmetic, and yes, it's mostly meant to illustrate a way of thinking about logical uncertainty. Basically, the question of logical uncertainty gets more interesting when you try to say "I have one particular model of PA in mind, but I can't compute which one; what should my prior be?"
Typo fixed; thanks.
I agree.
I see these desiderata more as a litmus test. I expect the approximation algorithm and the prior it's approximating will have to be developed hand-in-hand; these desiderata provide good litmus tests for whether an idea is worth looking into. I tend to think about the problem by looking for desirable limits with approximation in mind, but I agree that you could also attack the problem from the other direction.