02 December 2007

Recitation instructors, TV pundits, and Poincare

I've been reading through the archives of Eliezer Yudkoswsky's Overcoming Bias, and I found an interesting posts:

Focus Your Uncertainty talks about the plight of a novice TV pundit who has to get what the market will do in advance, in order to decide how ey will allocate time to preparing remarks, so that ey can explain what happened after the fact.

I face much the same problem in teaching recitations for calculus classes. My recitations are very much driven by the students, in that I spend most of the time answering their questions. Which questions do I prepare for ahead of time, and which do I just decide I'll figure out on the fly? Here there is another wrinkle -- there are some questions that students aren't that likely to ask about, but which will take a long time to prepare.

In the case of the TV pundit, I'd almost say that having a long explanation for a certain action in the market is in itself evidence for that action being unlikely, basically by Occam's razor. This doesn't carry over to the calculus classes -- the things having long explanations (that is, the hard problems) are exactly the ones the students are most likely to ask about.

Yudkowsky writes, in that post:
Alas, no one can possibly foresee the future. What are you to do? You certainly can't use "probabilities". We all know from school that "probabilities" are little numbers that appear next to a word problem, and there aren't any little numbers here. Worse, you feel uncertain. You don't remember feeling uncertain while you were manipulating the little numbers in word problems. College classes teaching math are nice clean places, therefore math itself can't apply to life situations that aren't nice and clean. You wouldn't want to inappropriately transfer thinking skills from one context to another. Clearly, this is not a matter for "probabilities".

That's something to keep in mind -- in working with probability one doesn't feel uncertain. Sometimes I feel the same way, and this may be because we've thoroughly axiomatized away the uncertainty. I was recently reading Poincare's book Science and Hypothesis(*), which includes a chapter on "the calculus of probabilities" -- a lot more uncertainty seems to permeate this chapter than would a similar chapter written now, because Poincare lived before the Kolmogorov axioms. But this is an interesting philosophical fact about probability -- we are saying things about uncertainty, but we know that they're true. And sometimes, as with the "probabilistic method" in combinatorics, this allows us to prove things about structures that don't involve uncertainty. I leave this to the philosophers.

(*) I was proud of myself for picking this up at a used bookstore for $6.95 -- but Amazon's selling it for $6.69! (Of course, I ought to not be too proud, since Penn's library almost certainly has it.)

1 comment:

Anonymous said...

Unequivocally, ideal answer