[extropy-chat] Paradox? What paradox?
gts
gts_2000 at yahoo.com
Sat Jan 6 22:01:31 UTC 2007
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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