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Janos2 comments on 0 And 1 Are Not Probabilities - Less Wrong

34 Post author: Eliezer_Yudkowsky 10 January 2008 06:58AM

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Comment author: Janos2 10 January 2008 06:49:09PM 1 point [-]

I suspect Eliezer would object to my post claiming that I'm confusing map and territory, but I don't think that's fair. If there's a map you're trying to use all over the place (and you do seem to), then I claim it makes no sense to put a little region on the map labelled "maybe this map doesn't make any sense at all". If the map seems to make sense and you're still following it for everything, you'll have to ignore that region anyway. So is it really reasonable to claim that "the probability that probability makes sense is <1"?


Measure theory gives a clear answer to this: it's 0. Which is fine. For all x, the probability that your rv will take the value x is 0. Actually the probability that your rv is computable is also 0. (Computable numbers are the largest countable class I know of.) What's false is the tempting statement that probability 0 events are impossible. It's only the converse that's true: impossible events have probability 0. There's another tempting statement that's false, namely the statement that if S is an arbitrary collection of disjoint events, the probability of one of them happening is the sum of the probabilities of each one happening. Instead, this only holds for countable sets S. This is part of the definition of a measure.