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srn347 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: [deleted] 04 January 2013 08:32:14AM 0 points [-]

Forgive me if this sounds condescending, but isn't saying "0 and 1 are not probabilities because they won't let you update your knowledge" basically the same as saying "you can't know something because knowing makes you unable to learn"? If we assign tautologies as having probability 1, then anything reducible to a tautology should have probability 1 (and similarly, all contradictions and things reducible to contradictions should have probability 0). For any arbitrarily large N, if you put 2 apples next to 2 apples and repeat the test N times, you'll get 4 apples N out of N times, no less (discounting molecular breakdowns in the apples or other possible interferences).

Comment author: Qiaochu_Yuan 04 January 2013 10:09:44AM -1 points [-]

You shouldn't assign tautologies probability 1 either because your notion of what a tautology is might be a hallucination.

Comment author: RichardKennaway 04 January 2013 11:02:07AM *  2 points [-]

This confuses object level and meta level. In probability theory, P(-A|A) = 0 and P(A|A) = 1, however uncertain you may be about Cox's theorem, or about whether you are actually thinking about the same A each time it appears in those formulas. No-one, as far as I know, has ever constructed a theory of probability in which these are assigned anything else but 0 and 1. That is not to say that it cannot be done, only that it has not been done. Until that is done, 0 and 1 are probabilities.

The title of the article is a rhetorical flourish to convey the idea elaborated in its body, that to assert a probability, as a measure of belief, of 0 or 1 is to assert that no possible evidence could update that belief, that 0 and 1 are probabilities that you should not find yourself assigning to matters about which there could be any real dispute, and to suggest odds ratios or their logarithms as a better concept when dealing with practical matters associated with very low or very high probabilities. There is a very large difference between saying that the probability of winning a lottery is tiny and saying that it cannot happen at all; with enough participants it is almost certain to happen to someone. That difference is made clear by the log-odds scale, which puts the chance of a lottery ticket at 60 or more decibels below zero, not infinitely far below. In a world with 7 billion people, billion-to-1 chances happen every day.

As an example of even tinier probabilities which are still detectably different from zero, consider a typical computer. A billion transistors in its CPU, clocked a billion times a second, running for a conveniently round length of time, a million seconds, which is about 12 days. Computers these days can easily do that without a single hardware error, which means that for every one of a million billion billion switching events, a transistor opened or closed exactly as designed. A million billion billion is about 1.5 times Avogadro's number. The corresponding log-odds is -240 decibels. And yet hardware glitches can still happen.

And P(A|A) is still 1, not any finite number of decibels.