Wednesday, January 2, 2013

More on qualitative probability and regularity

For probabilities that have values, regularity is normally defined as the claim that P(A)>0 whenever A is non-empty. Given finite additivity, this entails that if A is a proper subset of B, then P(A)<P(B). If instead of assigning probability values we deal with qualitative probabilities—i.e., probability comparisons—we now have a choice of how to define regularity:

  • the probability comparison ≤ is weakly regular provided that ∅<A whenever A is non-empty
  • the probability comparison ≤ is strongly regular provided that A<B whenever A is a proper subset of B.

If one assumes the axiom:

  • (Additivity, de Finetti) If AC=∅ and BC=∅, then AB if and only if ACBC,
then weak regularity entails strong regularity.

In a recent post, I showed that there is no rotationally invariant strongly regular qualitative probability defined on all countable subsets of the circle. But perhaps there is a useful weakly regular one that does not satisfy Additivity?

I don't know. Here's a start. Let P be the Bernstein-Wattenberg hyperreal-valued measure on the circle. Define AB if and only if for some rotations r and s we have P(rA)≤P(sB). Define AB if and only if not B<A. Then ≤ is weakly regular, and rotationally invariant. But I can't prove that it's transitive. Is it?

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