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MTGandP comments on Original Research on Less Wrong - Less Wrong
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Comments (47)
Thanks. Sometimes I learn a lot from people saying fairly-obvious (in retrospect) things.
In case anyone is curious about this, I guess that Eliezer knew it instantly because each additional data point brings with it a constant amount of information. The log of a probability is the information it contains, so an event with probability .001 has 2.3 times the information of an event of probability .05.
If that's not intuitive, consider that p=.05 means that you have a .05 chance of seeing the effect by statistical fluke (assuming there's no real effect present). If your sample size is n times as large, the probability becomes (.05)^n. (Edit: see comments below) To solve
(.05)^n = .001
take logs of both sides and divide to get
n = log(.001)/log(.05)
Why?
You mean why isn't the information of a particular number just its length, or its size, and is its log of all things?
Because you can think of each part of the number as telling you how to navigate a binary tree to the node target meaning, and the opposite of a binary tree is the logarithm; at least, that's how I think of it when I use it in my essays like Death Note anonymity.