[extropy-chat] Paradox? What paradox?

[gts]

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