[extropy-chat] Paradox? What paradox?

[Benjamin Goertzel]

Trying again, my previous attempt to send this message resulted in it
getting bounced back as SPAM ...

ben g

On 1/6/07, Benjamin Goertzel <ben at goertzel.org> wrote:
> Von Mises' wine/water paradox is fairly convincingly addressed in
>
> http://philsci-archive.pitt.edu/archive/00002487/01/Indifference_new_...Burock_2005.pdf
>
> where it is shown that the paradox goes away if one assumes a 2D
> sample space covering the set of pair of values
>
> (wine/water, water/wine)
>
> as is appropriate since these two ratios are dependent quantities.
>
> This suggests that the Principle of Indifference is tricky but perhaps
> not completely mistaken.
>
> Paradoxes involving the PI are derived by assuming multiple
> contradictory sample spaces for the same problem, whereas it is only
> legit to compare to probabilities if they are derived within the same
> sample space.
>
> -- Ben G
>
> On 1/6/07, gts <gts_2000 at yahoo.com> wrote:
> > Here is yet another paradoxical consequence of the Principle of
> > Indifference.
> >
> > (The Principle of Indifference is a principle of epistemic
> > [non-objectivist] probability theory, which states that if each of n
> > possibilities are indistinguishable except for their names -- that is if
> > we have no reason to expect one possibility more than another -- then each
> > possibility should be assigned a probability equal to 1/n.)
> >
> > These paradoxes demonstrate what happens when we apply the principle to
> > continuous variables.
> >
> > Suppose we have a mixture of wine and water and we know that at most there
> > is 3 times as much of one as the other, but nothing about the mixture. We
> > have:
> >
> > (1/3) is less than or equal to (wine/water) is less than or equal to (3)
> >
> > and by the Principle of Indifference, the ratio of wine to water has a
> > uniform probability density in the interval [1/3, 3]. Therefore...
> >
> > Probability that wine/water is less than or equal to 2 = (2 - 1/3)/(3-1/3)
> > = 5/8
> >
> > But also...
> >
> > (1/3) is less than or equal to (wine/water) is less than or equal to (3)
> >
> > And by the Principle of Indifference, the ratio of water to wine has a
> > uniform probability density in the interval [1/3, 3]. Therefore...
> >
> > Probability that water/wine is greater than or equal to 1/2 = (3 -
> > 1/2)/(3-1/3) = 15/16
> >
> > But the events (wine/water is less than or equal to 2) and (water/wine is
> > greater than or equal to 1/2) are the same, and the Principle of
> > Indifference gives them different probabilities.*
> >
> > Moral of the story: the Principle of Indifference is at best problematic
> > and at worst completely mistaken, at least with respect to continuous
> > variables.
> >
> > -gts
> >
> > *Gillies, 2000
> >
> >
> >
> >
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> >
>