Sniffnoy comments on Beyond Bayesians and Frequentists - Less Wrong

36 Post author: jsteinhardt 31 October 2012 07:03AM

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Comment author: Eliezer_Yudkowsky 31 October 2012 07:57:19AM 11 points [-]

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/

Comment author: Sniffnoy 31 October 2012 10:03:46PM 3 points [-]

Well, Cox's Theorem still assumes that you're representing belief-strengths with real numbers in the first place. Really you should go back to Savage's Theorem... :)