From: Ben Goertzel (ben@goertzel.org)
Date: Fri Sep 16 2005 - 04:36:34 MDT
Eli,
Yeah, I already realized that if one makes various independence assumptions
(like you do in your assumption 2) then the Hempel paradox doesn't arise ...
Also, it's clear that from the point of view of the experience of actual
organisms, "choosing a random non-black object" is not something that can
ever actually be done...
However, these observations don't really get at the essence of the matter,
in my view.
-- Ben
> Ben Goertzel wrote:
> > Just one more thing...
> >
> > I started out this whole silly thread by saying that:
> >
> >>>If probability theory as standardly deployed states that an observation
> >>>of a non-black non-raven provides a NON-ZERO amount of evidence toward
> >>>the hypothesis that all ravens are black, then this shows there is
> >>>something wrong with probability theory as standardly deployed.
> >
> > I admit that in my followup discussions, after making this statement,
> > I manifestly failed to demonstrate its truth...
>
> Ben, standard probability theory says no such thing. You have to
> provide a number of additional assumptions before probability theory,
> the logic of science, makes any logical statements about the
> probabilities involved. Observe a green lampshade one way, it means one
> thing, observe it with different prior information, it means
> something else.
>
> > Instead, I made some careless and silly errors, both with the standard
> > formulation of probability theory and with my own PTL formulation. I
> > apologize for this -- I'm not usually quite *that* error-prone even
> > when badly overworked, but what can I say, it happens from time to
> > time....
> >
> > However, after all that, I *still* hold the same intuition that I had
> > originally. And this is with the probabilistic arguments regarding the
> > Hempel paradox quite fresh in my mind and quite fully understood both
> > conceptually and arithmetically.
> >
> > I don't doubt the math of probability theory, but I still have a nagging
> > intuitive suspicion that the way the math is being applied to
> this situation
> > is not conceptually right. Furthermore, I still have the same suspicion
> > that this conceptual wrongness is related to other problematic issues
> > with standard AI deployments of probability theory such as Bayes nets.
>
> Ben, did I wish to justify your intuition, I would use the following
> reasonable-sounding assumptions.
>
> 1) Random sampling of objects (not random sampling of a
> non-black object).
>
> 2) No correlation between your belief in the color mixture of ravens
> and your belief in the number of ravens. That is, whether some ravens
> are non-black does not, so far as you know, have anything to do with the
> total number of ravens of all colors, nor the proportion of non-ravens
> to ravens. (For all you know there might be a correlation, but if so,
> you have no idea what direction it's in.) I.e., if you think that half
> of all ravens are white, this doesn't change your estimation of how many
> ravens there are in the world.
>
> 3) The only thing you know about the result is that it was not a raven.
>
> Now if you observe an object randomly selected from the set of all
> objects, and you know only that it is not a raven, this says nothing
> about the color mix of ravens, because your color mix hypothesis classes
> all assign equal likelihood to your seeing a non-raven (of
> whatever color).
>
> But that's not what probability theory says, as such.
>
> Probability theory just says to further specify your assumptions.
>
> Probability theory has plenty of room to encompass structure of the sort
> Russell Wallace refers to, if you know more about the object than that
> it is not a raven.
>
> And if you randomly sample a non-black object, and the only thing you
> know about the result is that the result is not "Raven", you are
> essentially stuck with this increasing the probability of the hypothesis
> "All ravens are black" under any reasonable prior assumptions. If you
> sampled a nonblack object and it was a raven, the probability of "All
> ravens are black" would go (way) down. So if you get the converse
> result of nonraven, the probability "All ravens are black" goes up.
> Unless you are a priori certain that the result will not be a raven
> because ravens are too rare. p(A) = p(A|B)p(B) + p(A|~B)p(~B). If
> p(A|B) > p(A) then p(A|~B) < p(A) if both probabilities are defined.
>
> --
> Eliezer S. Yudkowsky http://intelligence.org/
> Research Fellow, Singularity Institute for Artificial Intelligence
>
>
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