Dean Foster's Research
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
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
- "An Information Theoretic Comparison of Model Selection," with
Bob Stine .
- "Variable selection in data mining: Building a predictive model for
bankruptcy," with Bob Stine , (talk).
- "Empirical Bayes Variable Selection,
(pdf)" with Edward George,
Biometrika, 2000, 731 - 747.
- "Local Asympotic Coding for Model
Selection (pdf)," with Bob Stine , , IEEE
Transaction on Information Theory, (1999) 1289 - 1293.
Confidence Intervals for the Error Variance in Stepwise
Bob Stine .
- "Universal Codes for Finite
Sequences of Integers Drawn from a Monotone Distribution", with
R. Stine and A.J. Wyner, To Appear in the IEEE Transactions on
Information Theory, 2002.
variable selection with Bayesian oracles. " with Bob Stine.
Sprinkling in a bit of game theory, led to considering a market for
variables. The idea is a player buys the oppertunity to try a
variable. If that variable turns out to be significant, then the
player is rewarded. The price can be paid in "alpha" or in "bits."
These ideas have lead to the following papers:
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.
- "Prediction in the Worst Case,"(JSTOR) The Annals of
Statistics, 19, (1991), 1084 - 1090.
- "A Randomization Rule for Selecting Forecasts," with Rakesh
Vohra, Operations Research, 41,
(1993), 704 - 709, with discussion by R. Clemen.
- "Asymptotic Calibration," with Rakesh
- "Calibrated Learning and
Correlated Equilibrium," with Rakesh
Vohra to appear in Games and Economic Behaviour.
- "Introduction to the Special Issue," (in honor of David
Blackwell) with R. Vohra, D. Levine, Games and Economic
Behavior, (1999), 1 - 7.
- "Regret in the On-line Decision
Problem," with Rakesh
Vohra, Games and Economic Behavior, 1999, 7 - 36.
- "A proof of
Calibration via Blackwell's Approchability Theorem.", Games and
Economic Behavior, 1999, 73 - 79.
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
- "Learning, Hypothesis Testing, and Nash
Equilibrium (ps)" (.pdf), with H. P. Young.
- "Learning with Hazy Beliefs," (1997) with H. P. Young. (No longer
current. See "Hypothesis testing" above.)
- "On the Impossibility of
Predicting the Behavior of Rational Agents (ps)," (.pdf) with H.P. Young,
- "On the Nonconvergence of Fictitious Play in Coordination Games," with
H. P. Young, Games and Economic Behaviour, 1998, 79 - 96.
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.
Record Asymptotics for Rolling Sample Variance Estimators," with
, 64, (1996), 139
- "Filtering and Forecasting with Misspecified ARCH Models: Making
the Right Forecast with the Wrong Model," with D. Nelson, Journal of
Econometrics, 67, (1995), 303 - 335.
- "Asymptotic Filtering Theory for Univariate ARCH Models,"(JSTOR) with D.
62, (1994), 1 - 41.
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
- "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