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shiney comments on Results from MIRI's December workshop - Less Wrong
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Comments (43)
Can't you somewhat patch Demski's proposal by sampling uniformly from S rather than doing it biased by length. That would generate the right probabilities for the 90% issue, provided that all the ϕ(n) are in S to start with. If not all the sentences were in S then there still be a bias towards ϕ(n) being true but it would be only for the ones such that ϕ(n) is in S and it would be lower.
This doesn't prefer simpler theories about the world, which "defeats the whole point" (to an extent): it can't be used as a general theory of induction and logical uncertainty.
Maybe I missed something but I could never see why there was anything intrinsically good about (say) the short bias in the Solomonoff prior, it seemed like the whole thinking bigger programs were less likely was just a necessary trick to make the infinite sums finite. If we consider the formalism in the technical note that only keeps a finite number of sentences in memory then we don't have to worry about this issue and can sample uniformly rather than (arbirtrarily?) picking a bias.
In your paper http://ict.usc.edu/pubs/Logical%20Prior%20Probability.pdf you have to worry about the infinite series summing to infinity as the probability distribution is over all sentences in a language, so you have to include this bias.
If we can avoid assuming simplicity is more likely and treat it as a fact about the world which we might learn after choosing a prior isn't that better?
I feel that the bigger issue with this proposal is that it doesn't solve the problem it's trying to solve as statements in S are biased towards being picked.