Monday, October 6, 2008

Happy Happy Joy Joy

I am one happy blogger. I have completely sorted out a little mathematical problem that's been plaguing me for ages. It's one where I already knew the result, but all the proofs I could construct were either too embarrassingly clunky to use (I mean, really, really awful), or elegant but handwavy in one place.

I realized last week we really needed this result for something I'm working on with my research student. I sat and thought about it for a while today and finally noticed that the one remaining bit of argument we needed to make was obvious if you just recast the whole problem as a count from a thinned Poisson process. The crazy thing was my old handwavy argument was in effect already doing that, I just had failed to recognize it for what it was. Now that it's been set up in the right way, all the handwavy aspects drop away, and a nice clean half-page argument based on already-known results establishes the result we need.

This is one of those moments when after the fact everything is so obvious that I feel inadequate for not having seen it much earlier, but for the moment the joy is undiminished, because what this small step gets us to is something dramatic (assuming showing a bunch of well-known-in-their-application-area people that what they've been saying and doing is completely wrong is dramatic).

1 comment:

Blake Stacey said...

Heh. It looks like I "invented" thinned Poisson processes a couple years ago while playing with a toy model of the fossil record to see how spurious "accelerating rates of change" could arise. Now I need to see how inept my attempts were by comparison. . . .