**From:** Ben Goertzel (*ben@goertzel.org*)

**Date:** Thu Sep 15 2005 - 06:38:13 MDT

**Next message:**Eliezer S. Yudkowsky: "Re: Hempel's paradox redux"**Previous message:**Ben Goertzel: "RE: Hempel's paradox redux"**In reply to:**Marcello Mathias Herreshoff: "Re: Hempel's paradox redux"**Next in thread:**Ben Goertzel: "RE: Hempel's paradox redux"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

Yes, this is the same as the "two coins" analogy, it's just more fully

formalized...

ben

*> This from http://www.goertzel.org/new_essays/hempel.htm :
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*> >> Given only this knowledge, the probability that the bag is in
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*> State 2 would
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*> >> seem to be
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*> >> b_2 = a_2 / (a_2 + a_4)
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*> >> and the probability that the bag is in State 4 would seem to be
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*> >> b_4 = a_4 / (a_2+ a_4)
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*>
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*> It might at first sight seem to be true, but it isn't.
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*> As a simplification let us only consider states two and four.
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*>
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*> Raven Non-Raven
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*> State 2: B W
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*> State 4: W W
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*>
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*> Let H be a random variable with value 2 or 4 for which state we are in.
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*> Let E be the random variable with value r (the midget retrieves a
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*> Raven) or n
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*> (the midget retrieves a non-raven.)
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*>
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*> That is:
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*> a_2 = P(H=2)
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*> a_4 = P(H=4)
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*> b_2 = P(H=2|E=n)
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*> b_4 = P(H=4|E=n)
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*>
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*> We want to calculate:
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*> p(H=2|E=r) and p(H=4|E=n)
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*>
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*> Now apply Bayes law:
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*> p(H=2|E=n) = P(H=2)*P(E=n|H=2)/P(E=n)
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*> p(H=4|E=n) = P(H=4)*P(E=n|H=4)/P(E=n)
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*>
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*> Now we can divide to get the odds ratio:
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*> b_2 / b_4 = (a_2*P(E=n|H=2))/(a_4*P(E=n|H=4))
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*>
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*> Now Ben would be completely right about his assumption if:
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*> P(E=n|H=2) = P(E=n|H=4)
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*>
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*> However, this is not true:
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*> P(E=n|H=2) = 1
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*> P(E=n|H=4) = 1/2 (because there is another non-black object around for the
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*> midget to stumble on, namely the raven)
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*>
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*> Instead we get:
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*> b_2/b_4 = 2*a_2/a_4
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*>
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*> In short, the odds ratio for all ravens being black has been
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*> increased by a
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*> factor of two when our midget procures a non-black object and it
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*> turns out to
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*> not be a raven. This most definitely is evidence that all ravens
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*> are black!
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*>
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*> -=+Marcello Mathias Herreshoff
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*>
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*>
*

**Next message:**Eliezer S. Yudkowsky: "Re: Hempel's paradox redux"**Previous message:**Ben Goertzel: "RE: Hempel's paradox redux"**In reply to:**Marcello Mathias Herreshoff: "Re: Hempel's paradox redux"**Next in thread:**Ben Goertzel: "RE: Hempel's paradox redux"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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