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cousin_it comments on Maximize Worst Case Bayes Score - Less Wrong Discussion
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Comments (22)
Can you explain why a sequence of probability assignments, whose WCB converges to a limit, must have a pointwise converging subsequence (i.e. the probabilities assigned to all sentences converge simultaneously)?
It must have a subsequence S1 which converges for the first sentence (because the interval [0,1] is compact). This subsequence must itself have a subsequence S2 which converges in the second sentence, which must have a subsequence S3 which converges in the third sentence and so on.
The subsequence we want takes the first entry of S1, then the second entry of S2, then the third entry of S3, and so on. For every n, after the nth entry, this is a subsequence of S_n, so the probabilities of the nth sentence must converge.
Note, that all the probabilities converge, but they do not converge uniformly. At any given time, there will be some probabilities that are still way off. This is a common analysis trick. Converging simultaneously on countably many axes is no harder that converging simultaneously on finitely many axes. Let me know if I should clarify further.
Thanks! That's a good trick, I didn't know it.
Reading further, it seems like your definition of P3 in terms of P1 and P2 is indeterminate when P1(phi)=0 and P2(phi)=1. I assume this hole can be patched. (ETA: I'm being stupid, this can't happen if P1 and P2 both maximize WCB, because we can assign a truth value to phi that will make either P1 or P2 have negative infinity Bayes score.)
Otherwise the proof seems fine at first glance. Great work! This is exactly the kind of stuff I want to see on LW.
You could make a similar complaint about the proof of coherence too. Just observe that clearly something that maximizes WCB can only assign probability 1 to tautologies and can only assign probability 0 to contradictions, so that can never happen.
Thanks! I have actually been thinking along these lines for about a year. (Notice that the update function for both the proofs of uniqueness and coherence are generalizations of the strategy I argued for.) I consider doing MIRIx a success just for inspiring me to finally sit down and write stuff up.
That update move is nice in that it updates the Bayes score by the same amount in all models. What I would really like to show is that if you start with the constant 1/2 probability assignment, and just apply that update move whenever you observe that your probability assignment is incoherent, you will converge to the WCB maximizer. I think this would be nice because it converges to the answer in such a way that seems unbiased during the entire process.