Tuesday, October 30, 2012

Nonconglomerability and hyperreal-valued probability

Note added later: This follows by applying Theorems 3.1 and 3.2 of Schervish et al. to the standard part of P.

Suppose P is a finitely-additive hyperreal-valued probability on the natural numbers such that P assigns infinitesimal value to each natural number. While groggy from a cold plus baby-feeding at night, I've been trying to prove that P is nonconglomerable in the following strongish sense: there is a partition A1,A2,... of the naturals, an event E, and real numbers a<b such that P(E|Ai)<a for all i but P(E)>b. Thus, if we are trying to figure out if E is true, and we plan to observe the Ai, we know ahead of time that no matter which Ai we observe, our probability for E will go down. (This, of course, violates van Fraassen's reflection principle.) This is likely already known, but I couldn't find it on the Internet.

My proof sketch starts by dividing into two cases. Either the standard part of P takes on infinitely many values or not. If it takes on infinitely many values, then the standard part of P is a merely finitely additive measure that takes on infinitely many values, and so by the 1984 Schervish et al. theorem, it is nonconglomerable, and hence so is P in my strongish sense. If, on the other hand, the standard part of P takes on finitely many values, then P is a convex linear combination, with real coefficients, of a purely infinitesimal finitely additive signed measure and a finite number of indicator functions of free ultrafilters. And then with a bit of work one can construct a counterexample to conglomerability, I think. (I am not completely sure which extensions of the reals this works for, but it does work for the hyperreals.)

No comments: