## Tuesday, June 16, 2015

### Mixing and matching conditional probabilities

Given an unconditional probability function P, one can always (at least given the Axiom of Choice) extend to a full conditional probability function, or a Popper function, that allows one to assign values to P(A|B) even when P(B)=0. Typically, the extension is not unique. In fact, it turns out that there is no logical connection between the conditional probabilities P(A|B) for B a null set (a set of zero probability) and the unconditional probabilities.

What do I mean by saying there is no logical connection? Well, it turns out we can mix and match and the null-probability-condition parts of Popper functions with the other parts. Suppose that P and Q are two Popper functions defined on the same sets. Then we can define a frankenfunction by letting R(A|B)=P(A|B) when B is not a null set and R(A|B)=Q(A|B) when it is a null set. And this frankenfunction is a perfectly fine Popper function.

This is a problem. There is a complete disconnect between the null-probability-condition of the Popper function and the non-null-probability condition parts. (Another manifestation of this problem is the fact that in many cases we lack conglomerability.)