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BT_Uytya comments on Beyond Bayesians and Frequentists - Less Wrong
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I haven't read this in detail but one very quick comment: Cox's Theorem is a representation theorem showing that coherent belief states yield classical probabilities, it's not the same as the dutch-book theorem at all. E.g. if you want to represent probabilities using log odds, they can certain relate to each other coherently (since they're just transforms of classical probabilities), but Cox's Theorem will give you the classical probabilities right back out again. Jaynes cites a special case of Cox in PT:TLOS which is constructive at the price of assuming probabilities are twice differentiable, and I actually tried it with log odds and got the classical probabilities right back out - I remember being pretty impressed with that, and had this enlightenment experience wherein I went to seeing probability theory as a kind of relational structure in uncertainty.
I also quickly note that the worst-case scenario often amounts to making unfair assumptions about "randomization" wherein adversaries can always read the code of deterministic agents but non-deterministic agents have access to hidden sources of random numbers. E.g. http://lesswrong.com/lw/vq/the_weighted_majority_algorithm/
I've tried to do something similar with odds once, but the assumption about (AB|C) = F[(A|C), (B|AC)] made me give up.
Indeed, one can calculate O(AB|C) given O(A|C) and O(B|AC) but the formula isn't pretty. I've tried to derive that function but failed. It was not until I appealed to the fact that O(A)=P(A)/(1-P(A)) that I managed to infer this unnatural equation about O(AB|C), O(A|C) and O(B|AC).
And this use of classical probabilities, of course, completely defeats the point of getting classical probabilities from the odds via Cox's Theorem!
Did I miss something?
By the way, are there some other interesting natural rules of inference besides odds and log odds which are isomorphic to the rules of probability theory? (Judea Pearl mentioned something about MYCIN certainty factor, but I was unable to find any details)
EDIT: You can view the CF combination rules here, but I find it very difficult to digest. Also, what about initial assignment of certainty?
EDIT2: Nevermind, I found an adequate summary ( http://www.idi.ntnu.no/~ksys/NOTES/CF-model.html ) of the model and pdf ( http://uai.sis.pitt.edu/papers/85/p9-heckerman.pdf ) about probabilistic interpretations of CF. It seems to be an interesting example of not-obviously-Bayesian system of inference, but it's not exactly an example you would give to illustrate the point of Cox's theorem.