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JoshuaZ comments on Naming the Highest Virtue of Epistemic Rationality - Less Wrong Discussion
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Comments (28)
What assumptions are you using here? In the simplest form this is obviously false. Simply let the Score function always be zero. Moreover, any Score function that satisfies this identity can be scaled by any number and still satisfy it. So not only does your log work but a log to any other base will work. Also if I believe in the axiom of choice (I think I need choice here to get a transcendence basis. Can someone more foundationally oriented comment if this is correct?), then there is a function f on the positive reals such that f(ab)=f(a)+f(b) and f(x) is not equal to a constant times log x. So one could just as well use that f as your score function.
I think your statement is true if you want your score function to be continuous and normalized so that one has Score(1/2)= -1.
On a completely different note, the repeated references to the Sequences come off as a bit off. You assume a high degree of detailed familiarity with the various essays that is unjustified. For example, the quote from the Twelve Virtues sounded familiar, but I almost certainly would not have been able to place it. Moreover, the way these quotes are used almost feels like you are quoting religious proof texts or writing a highschool English essay rather than actually using them in a useful way.
Twelve virtues is really popular, that's why I wrote that. I'll take it out if it's distracting. (done and done)
I was actually disagreeing with them, you understand this right. If not let me know, cause that's important for my readers to get off the bat. Those are the two quotes I remember most clearly saying that we shouldn't name the highest value. i put them there as evidence that not naming the highest value is standard LW doctrine. So if it came off as quoting doctrine perhaps I made my point.
On the Log stuff. I know other bases work fine too. But normalization is nice, actually i think I write log_b somewhere.
Not just other bases. I can construct another function as follows: Fix a basis for R over Q. I can do this if I believe in the axiom of choice. Call the elements of that basis x(i). Consider then the function that takes elements of log_x, writes them with respect to the basis and then zeros the coordinate connected to a fixed basis element x(0). This function will have your desired property and is not a constant times log.
Interesting. Is it continuous as well?
I may be wrong. But I think EY say's in tech explanation that no other function satisfies that condition and is also proper.
Is this f a proper scoring rule?
No. This is wildly non-continuous. It also isn't proper. This is why specifying what your hypotheses are for your theorems is important.
good point.But I think I said it had to be proper. I've made that more explicit.