[extropy-chat] what is probability?
Benjamin Goertzel
ben at goertzel.org
Fri Jan 12 16:26:26 UTC 2007
Yes, but in terms of the subjective theory of probability, it seems
that certain ways of using the PI are "incoherent" whereas others are
"coherent." The paper I referenced described coherent ways of using
PI in the wine/water situation and other commonly discussed
situations...
A rational mind may, at a certain point, not be able to tell which way
of using the PI is going to be coherent in the context of future
observations. But it should be able to tell which ways will be
coherent in the context of prior observations, if it has adequate
computational resources to do the calculations.
ben
On 1/12/07, gts <gts_2000 at yahoo.com> wrote:
> "The Ramsey-De Finetti theorem is a remarkable
> achievement, and clearly demonstrates the superiority
> of the subjective to the logical theory. Whereas in
> the logical theory the axioms of probability could
> only be justified by a vague and unsatisfactory appeal
> to intuition, in the subjective theory they can be
> proved rigorously from the eminently plausible
> condition of coherence [...] In addition, the
> subjective theory solves the paradoxes of the
> Principle of Indifference by, in effect, making the
> principle unnecessary, or at most a heuristic device."
>
>
> -D. Gillies, (2000) _Philosophical Theories of
> Probability_: Routledge
>
> Concerning the paper cited by Ben in which someone
> named Durack (if I have his name correctly) proposed a
> solution to the wine/water paradox, I've been thinking
> about that (always a bad sign)...
>
> The author proposes a reasonable solution to the
> wine/water paradox, but I've been thinking about the
> general applicability of his idea to the bertrand
> paradoxes.
>
> The main idea in that paper is that paradoxes of the
> Principle of Indifference (the PI) can be avoided if
> we ask the right questions. Specifically the author
> makes a persuasive argument that the probability of an
> outcome is always conditional on the sample space that
> contains it, and that the PI paradoxes go away if only
> information about the relevant sample space is made
> explicit in the questions that lead to the supposed
> paradoxes.
>
> But something about that idea just seems too obvious..
> and it occurs to me why: it is obvious because it's
> nothing new at all, but rather just a sort of
> restatement of the classical theory of Pascal, Fermat
> and Laplace.
>
> The basic problem still remains in the case of, for
> example, selecting the random cube. What is the
> probability of the event in question? We're inclined,
> after reading that paper, to reply, "The answer
> depends on our choice of sample space, information not
> included in the question." And of course there is
> nothing at all wrong with our reply.
>
> But here is the rub:
>
> The principle of indifference exists as a supposed
> logical principle for us to use in exactly this type
> of situation of high uncertainty! To answer that "more
> information is needed" is to admit that the PI has
> failed us.
>
> Real-world situations can arise in which probability
> decisions must be made with little or no knowledge of
> the distribution or sample space, situations analogous
> to that which we face in the aforementioned paradox of
> selecting a random cube from the output of a cube
> factory. It seems natural to invoke the principle of
> indifference in such situations - it's certainly not
> *incorrect* use it as a way to determine Bayesian
> priors - but somehow I think it is nevertheless
> incorrect to imagine that our decision to invoke the
> PI is always strictly *rational*.
>
> So I'm finding myself back in agreement here with
> Gillies above, that the PI is "at most a heuristic
> device"...
>
> -gts
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