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JGWeissman comments on Take heed, for it is a trap - Less Wrong
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Comments (187)
Zed, you have earned an upvote (and several more mental ones) from me for this display of understanding on a level of abstraction even beyond what some LW readers are comfortable with, as witnessed by other comments. How prescient indeed was Bayesian Bob's remark:
You can be assured that poor Rational Rian has no chance when even Less Wrong has trouble!
But yes, this is of course completely correct. 50% is the probability of total ignorance -- including ignorance of how many possibilities are in the hypothesis space. Probability measures how much information you have, and 50% represents a "score" of zero. (How do you calculate the "score", you ask? It's the logarithm of the odds ratio. Why should that be chosen as the score? Because it makes updating additive: when you see evidence, you update your score by adding to it the number of bits of evidence you see.)
Of course, we almost never reach this level of ignorance in practice, which makes this the type of abstract academic point that people all-too-characteristically have trouble with. The step of calculating the complexity of a hypothesis seems "automatic", so much so that it's easy to forget that there is a step there.
If P is the probability that an ideal Bayesian would assign to a proposition A on hearing A but having observed no relevant evidence, then you have described the meta expected value of P in logical ignorance before doing any calculations (and assuming an ignorance prior on the distribution of propositions one might hear about). It seems to me that you have made excessively harsh criticism against those who have made correct statements about P itself.
See my other comments. In my opinion, the correct point of view is that P is a variable (or, if you prefer, a two-argument function); the "correct" statements are about a different value of P from the relevant one (resp. depend on inappropriately fixing one of the two arguments).
EDIT: Also, I think this is the level on which Bayesian Bob was thinking, and the critical comments weren't taking this into account and were assuming a basic error was being made (just like Rational Rian).