Friday, September 14, 2012

A new sceptical argument

Let's say that an infinite sequence of real-numbered observations is generated by independent runs of a random process. Suppose that we can represent the runs of the random process as independent and identically distributed random variables X1,X2,.... Recall that a random variable is a function f from some probability space Ω to the reals R with the measurability property that f−1[B] is a measurable subset of Ω for every Borel-measurable subset B of the reals R (and it's enough to check this for B an interval, since the intervals generate the Borel sets).

It turns out that under these assumptions we can almost surely recover the distribution of the random variables Xi from the observed sequence. For almost surely the frequency of the observations fitting into any given interval with rational numbers as ends will converge to the probability that Xi is in that interval. And since there are only countably many such intervals, almost surely for every such interval I we can read off the probability P(Xi in I) from the observed frequencies. And then by the uniqueness condition in the Caratheodory extension theorem, we can recover the probability of Xi being in A for any Borel subset, not just a rational-ended interval.

So far this sounds like a kind of vindication of infinitary frequentism. It is a helpful, optimistic result.

But notice a crucial assumption the recovery of the distribution of the Xi made: that the Xi are measurable when considered as functions to the Borel-measure space R. But there are infinitely many other σ-algebras on R besides the Borel one. When recovering the distribution from the observations, what justifies the assumption that the Xi are measurable as a function to R considered as coming with the Borel σ-algebra?

We might have some hope that the recovery process will be likely to fail if the Xi aren't measurable in this way, so that if the recovery process succeeds, we have good reason to think that the Xi have this measurability condition. But I suspect that some analogue of these results will show that this won't work.

So I guess we just need to assume measurability with respect to Borel sets. But why? Because God loves the Borel sets? It's not so crazy. I love the Borel sets, and God made me in his image, after all. :-)

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