# Dean Foster's Research

## Variable selection

Everyone does stepwise regression--statisticians don't approve, but then statisticians don't approve of many things. Edward and I wanted to justify stepwise regression and hopefully find a better way of doing it. Our justification was to consider a risk-ratio. We then optimized this criterion and came up with a proceedure that suggests adding variables if the F-statistic is bigger than 2 log p (basically this is Bonferroni). While we were waiting around for referee's reports we talked to Donoho and Johnstone who also had a paper under review that had the same idea.
The next big gain in variable selection was to switch from 2 log p to 2 log (p/q). This value is the smallest cut-off that doesn't obviuosly over fit. Decreasing it even a little bit will led to massive overfitting. So the question is, can we prove it doesn't over fit?

Edward and I have approached it from an empirical Bayes perspective. Bob and I have approached it from information theory. Neither approach has lead to a proof that the risk has desirable properties.

Bob Stine and I finally got a risk result for estimators of this type.

## Learning models (Calibration/No-regret/game theory)

Suppose you observe a sequence of events--but you are unwilling to consider a probabilistic model for these events. Can you still come up with good forecasts? Blackwell (1956) and Hannan (1957) showed that this was possible (as did I, but they were first). Rick's and my review paper (1999) discuss all the various people who have independently rediscoved these sorts of results.

In 1991 Rick and I also came up with a scheme that has a property called calibration. The idea of calibration is to make sure that your forecasts make empirical sense. So when you say there is a .2 chance of rain, it should rain empirically 1/5 of the time. This idea (along with our idea of no-regret) has a nice game theoretic applications: calibrated rules converge to correlated equilibria.

My work with Rick showed that one can learn correlated equilibria. But, can one learn the more traditional concept of a Nash equilibrium? Peyton and I extended Jordon's proof that Bayesians can't learn a Nash equilibrium to include Bayesians that do discounting.
It turns out that you can make a version of calibration that can be used as a coordinating device to discover a NE. It is still as exponentially slow as the exhaustive search done in the "Hypothesis Testing" approach.

## ARCH models

If you look the stock market, there are periods when not much is happening and the prices are stable and there are periods when the prices flucuate all over the place. Dan Nelson had me assist him in his analyis of these ARCH models.

## Affirmative action

Rick and I saw a talk by Judge Posner, that was so rabidly free-market that we wanted to write a piece showing that the free-market can't solve all problems. (Posner was argueing that rape is primarlly a search cost issue.) So we wrote this little model which showed that if all agents are rational, a free market is unable to remove arbitary and capricious discrimination. While our paper was under review, we were given a copy of a paper by Coate and Loury which had basically the identical model. Interestingly enough, it had the opposite conclusion.
• "An Economic Argument for Affirmative Action," with Rakesh Vohra, Rationality and Society, 4, (1992), 176 - 188, with discussion by G. Loury, by D. Friedman, and by J. Heckman and T. Philipson.

## Evolution and games

In 1988, Peyton Young and I came up with the idea of merging population genetics with game theory. This had be done for deterministic dynamics already--we just added noise to the existing definition. This is my most frequently cited paper.

• "Stochastic Evolutionary Games Dynamics," with H.P. Young, Journal of Theoretical Population Biology, 38, (1990), 219 - 232.
• "Cooperation in the Short and in the Long Run," with H.P. Young, Games and Economic Behavior, 3, (1991), 145 - 156.

## Other work in "progress"

One current project that has web pointers that I am working on is computer go. We are trying to use pattern matching to develop a go player.

I've currently only put up only one open source project called baby-lakos). It implements a way of doing levilization ala lakos. This was enough to get me certified as a apprentice on Advogato. Hopefully when the go progrect goes open-source I'll upgrade to journeyman.

Another project is the life calculator. Choong Tze Chua is now working on this project with us. He has developed a more complete version of the life calculator.