« Preseason Week 1 Stats Available and Other File-Related Errata | Main | Books I Recommend and More on Z-Scores »
August 16, 2008
Bayes' Theorem Braindump
Disclosure: this post is a braindump. Read at your own discretion!
Blog reader Kevin asked me a question about whether I've used the Bayes' Theorem in any of my calculations. I did not expect this question (and I should have paid more attention to Bayes' Rule in SI680). So - I went back and reviewed my knowledge. If you're not familiar with Bayes', I'd spend about 45 minutes and work through this intuitive explanation of Bayes. This question got the cogs in my brain turning. I just spent about an hour walking around Ann Arbor, trying to figure out a way to apply Bayes' to my dataset. I remain stumped. Here's my line of thinking though (not sure whether it's right or not):
1. For Bayes' to work, we need prior probabilities. 2. Then we need some conditional probabilities. 3. We can then generate posterior probabilities - revised priors.
Super. Unfortunately, none of my data is currently normalized as probabilities. Rather, my game-by-game data has been normalized as z-scores, which aren't really probabilities. We do have the predicted player-v-team score matrix - i.e. how John Kitna's past performance compares to other quarterbacks and how re-normalized to generate a prediction. Here's John Kitna's prediction entry:
+----------+------+------------+-------+--------+------+ | name | id | pid | gid | points | team | +----------+------+------------+-------+--------+------+ | J. Kitna | 5383 | 00-0009311 | 29529 | 11.45 | ATL | | J. Kitna | 5576 | 00-0009311 | 29546 | 11.34 | GB | | J. Kitna | 5799 | 00-0009311 | 29569 | 11.62 | SF | | J. Kitna | 6023 | 00-0009311 | 29591 | 9.70 | CHI | | J. Kitna | 6162 | 00-0009311 | 29606 | 11.86 | MIN | | J. Kitna | 6354 | 00-0009311 | 29625 | 13.10 | HOU | | J. Kitna | 6497 | 00-0009311 | 29634 | 10.81 | WAS | | J. Kitna | 6559 | 00-0009311 | 29645 | 9.70 | CHI | | J. Kitna | 6762 | 00-0009311 | 29661 | 10.26 | JAC | | J. Kitna | 6837 | 00-0009311 | 29674 | 10.30 | CAR | | J. Kitna | 7027 | 00-0009311 | 29693 | 8.69 | TB | | J. Kitna | 7146 | 00-0009311 | 29704 | 12.25 | TEN | | J. Kitna | 7342 | 00-0009311 | 29723 | 11.86 | MIN | | J. Kitna | 7545 | 00-0009311 | 29742 | 10.58 | IND | | J. Kitna | 7664 | 00-0009311 | 29755 | 13.16 | NO | | J. Kitna | 7829 | 00-0009311 | 29772 | 11.34 | GB | +----------+------+------------+-------+--------+------+
The predictions don't seem to be quite low and don't have enough variance. I'm going to have to look into that. Back to Bayes' - each of these predictions has a probability associated with it (or at least a range within 95% confidence intervals.) Can I then use Bayes' to generate a more accurate prediction on how Kitna does against Green Bay or New Orleans? I don't think so, well, I'm not sure. More research needed.
If you know how we can apply Bayesian rationality to these statistics, please reply and let me know (or at least point me in the right direction)
Posted by haydenth at August 16, 2008 10:15 PM