Sunday, June 26, 2011

Unconditional probabilities underdetermine conditional probabilities

Consider two random processes, R1 and R2. Process R1 randomly and uniformly chooses a real number between 0 and 1, with each number being a possible outcome. Process R2 works as follows. It starts with the result of R1. If the result of R1 is other than 1/2, that is also the result of R2. If the result of R1 is 1/2, then the result of R2 is defined to be 1/4.

Observe that for any measurable subset S of the interval between zero and one, the probability that R1 generates a number in S equals the probability that R2 generates a number in S. To see this, observe that unless R1 results in 1/2, R2 will result in the same thing as R1. So, the probability that R1 yields something in S differs from the probability that R2 yields something in S by at most the probability that R1 gives 1/2. But the probability that R1 yields 1/2 is zero, so the probability that R1 yields something in S does not differ from the probability that R2 yields something in S.

Now, intuitively the conditional probability that R1 results in 1/2 given that it results in either 1/2 or 3/4 is a half. But intuitively the conditional probability that R2 results in 1/2 given that it results in either 1/2 or 3/4 is zero, since R2 cannot result in 1/2. So, the conditional probabilities for R1 and R2 differ, while the unconditional ones are the same. So we cannot define conditional probabilities in terms of unconditional ones (and in particular, we cannot define conditional probabilities for zero-probability events using the Radon-Nikodym theorem or some other limiting procedure) in a way that does justice to our intuitions about conditional probability. This conclusion is of course the central point of Hajek's "What conditional probability could not be."

Of course, to me when wearing my probability theorist's hat, this doesn't matter. When I wear my probability theorist's hat, probability measures are simply measures with total probability one, and everything of significance is defined only up to almost sure equality (X=Y almost surely iff P(X=Y)=1) so the distinction between R1 and R2 is irrelevant. Such probability measures need not have any epistemological or physical interpretation. But there is a non-mathematical notion of conditional probability as well, one tied to epistemological and physical applications, and it is that kind of conditional probability that does not supervene on unconditional probability.

3 comments:

Alexander R Pruss said...

Analysis just accepted my paper based on a variant this little argument. Yay! It's my first paper in Analysis.

Marc Belcastro said...

The first of many, I suspect. =) Congratulations!

man with a computer said...

Congratulations, and merry Christmas!