Friday, May 11, 2012

Hausdorff Paradox and conditional probabilities

The Hausdorff Paradox shows that given the Axiom of Choice, there is no finitely additive probability measure defined for all subsets of the surface of a ball that is invariant under rotations—that assigns the same measure to a subset of the surface of the sphere and to a rotation about any axis through the center of that subset. Because of results like that, the standard Lebesgue measure on the surface of a ball is only defined for some subsets, i.e., the measurable ones.

Now, in classical probability, we can only define conditional probability when we condition on an event A with non-zero probability. We then use the formula P(B|A)=P(B&A)/P(A). Some have tried to come up with axiomatizations that allow for conditioning on all non-empty measurable sets, including zero-probability ones.

We should not hold our breath for success. Here's why. Let C be a solid ball of unit volume. Let P be Lebesgue measure on C: the measure of a subset of C is just the volume of the subset. We expect the Lebesgue measure on C to be invariant under rotations about all axes through the center of the ball. We would also expect that the conditional probability to be thus invariant. I.e., if r is a rotation about an axis through the center, we would expect it to be that P(r(B)|r(A))=P(B|A). Unfortunately this cannot be done, at least not if one assumes the axioms that P(−|A) is a finitely additive measure on the P-measurable subsets of C and P(A|A)=1. For let A be the surface of a ball concentric with C but of smaller radius. Then the volume of A is zero: A is a two-dimensional surface, after all. Moreover, Lebesgue measure has the property that every subset of a set with zero measure is also measurable (and has zero measure). So P(−|A) will be a finitely additive probability measure on all subsets of A. If we have our rotation invariance condition, then P(B|A)=P(r(B)|r(A))=P(r(B)|A) since r(A)=A (the sphere A is invariant under rotation through its center). So, P(−|A) will be a finitely additive rotation-invariant probability measure on all subsets of A, which violates the Hausdorff Paradox (assuming the Axiom of Choice).

Put it differently: Any conditional probability assignment that allows conditioning on all non-empty subsets will exhibit an unacceptable rotational bias.

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