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JoshuaZ comments on Naming the Highest Virtue of Epistemic Rationality - Less Wrong Discussion
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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.