Jason (jcreed) wrote,

Terrence Tao has a post about probability theory that made me puzzled at first, but I'm growing to like it more and more as I digest it.

There's an important insight to be had somewhere along the line that probability theory is "just" measure theory, that you can stop worrying about what words like probable "really mean" and get a significant amount of work done by just thinking of probable events as just "taking up more space" in the measure space of all possible worlds.

But Tao is making a further point, that even the set-theoretic underpinnings of measure theory are somehow excessively concrete. We don't have to go so far as to philosophically settle what randomness "really is", but we do benefit from building up a more first-class intuition for the concept of measure than it being just a map from sets to reals.

I like his point that extension (or what might also be called "refinement") of a measure space by observation of new experiments/events is the thing-which-concepts-must-be-robust-under for probability theory, just as change-of-coordinates is the thing-which-concepts-must-be-robust-under for linear algebra. Is there a category-theoretic way of saying this, I wonder? It's not immediately occurring to me, but I suspect I'ma feel dumb when it does, or when someone points it out.

The thing I'm getting stuck on is that change-of-coordinates (or homeomorphism, or algebraic isomorphism) are all, well, isomorphisms. They're two-sided, bidirectional, symmetric. Extension of probability spaces is directed, so it doesn't seem to make sense to say that the theory should quotient out by it.
Tags: math, probability
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