Faculty Reflections

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• Tom Cover
• David Donoho
• Brad Efron
• Confessions of a Scientific Dilettante (name withheld on request)

Tom Cover

Probability theory has always been sexy and exciting. The excitement comes from the intangibility of the subject of probability itself. There seems to be nothing physical there. On the other hand, some very concrete and deterministic theorems come out of probability theory.

For example, the average of a large number of the independent, identically distributed random variables has a deterministic limit equal to the expectation. This is the strong law of large numbers. There are some other deterministic gems in probability theory--the central limit theorem, the law of the iterated logarithm and the ergodic theorem. Can you imagine the excitement of seeing a formula like

coming out of thin air?

I was led to probability by my interest in gambling and poker. As a graduate student, I worked out the optimal raising and calling strategy in simple forms of poker. I also worked out the optimal doubling strategy in backgammon. Having these aces up my sleeve gave me the confidence to play as though I were the best. And confidence leads to good play. I was quite successful in poker as a graduate student. Later I heard about Thorp's then-unpublished work on beating blackjack by conditioning play on the distribution of the remaining cards in the deck. Another graduate student and I made a good deal of money following this strategy.

The power of probability comes up even at the simplest levels. For example, the Monty Hall paradox. One has gold behind one of three curtains and is allowed a chance to switch to one of the other two curtains after the game host has pulled aside a curtain without gold behind it. Surprisingly, you are always better off switching. This illustrates the principle of restricted choice. Also, Bertrand's paradox and a number of bar bets show the counterintuitive nature of probabilistic results even at the most primitive level. It is hard to believe that a subject which becomes counterintuitive so early will fail to be interesting.

What I'm working on now is mostly in the realm of information theory. That's another intangible concept. What is the information in an English sentence? Or in 100 flips of a bent coin? Everyone agrees that the quantity known as the Shannon entropy is the amount of bits of randomness in these various examples. I have spent the last ten or fifteen years trying to find out what the subject of information really is. To do this, I'm trying to identify the extreme points of the theory and unify them. In the process one can say much about some of the old questions like the second law of thermodynamics. Why does entropy increase and why does there seem to be an arrow of time?

Apparently, it is not true that entropy increases for every Markov chain. Nonetheless, in the physical world the second law of thermodynamics says that entropy always increases. Is our Markovian universe then necessarily restricted to a certain family of what we might call ``physical" Markov processes? I think not. I think something else is going on. But what?

Last year I had a chance with Professor Keller in the math department to create an undergraduate course called ``Mathematics and Sports.'' Here we tried to identify new strategies and new diagnostic statistics for all of the existing sports. Since sports statistics is one of my hobbies, I thought this would be an easy job. It wasn't. Nonetheless, I anticipate teaching a course like this again in two or three years.

Many years ago Herbert Robbins, one of the great creative minds in the field of mathematical statistics, was able to show that one can do better solving several independent statistical problems together rather than separately. This is a crazy idea because the problems are independent. Nonetheless, he was right. We tried to apply the same point of view to portfolio theory to argue that in the presence of a continuum of clever portfolio investment strategies, you can do as well as if you had known ahead of time which of the strategies was best. In particular, you can asymptotically outperform the best stock.

So the gist of all this is that there are a lot of simple and shocking statements--some at an elementary level, some at an advanced level--that come out of probability and statistics. It's the existence of these as yet unfound statements that drives my interest in the subject.
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David Donoho - Statistics: The Best-Kept Secret

When I make new acquaintances and say that I am a statistician, I sometimes observe surprise on their faces as though the existence of a field called "statistics" is totally new to them. It is certainly true that statistics has low visibility in the popular press.

Actually, statisticians are too busy having fun to worry much about publicizing themselves or what they do. In no other field can you be (a) part mathematician, (b) part computer hacker, (c) part scientist, and (d) part ethical conscience of the whole world.

Roles (a)-(d) (and a few others!), are so fulfilling that Public Relations seems pretty uninteresting by comparison. As a result, statistics is a near-invisible profession, in the popular mind, and perhaps also in the undergraduate mind.

This trend is only going to intensify. Computerization, telecommunications, terabyte-capacity personal computer data storage, refinements of scientific instruments: these are massive forces in the world at large which are creating massive new data bases, and new types of data analysis problems. Soon statisticians will be so much in demand that just deciding which projects are most interesting will take up a lot of their time. Already, a statistician has difficult choices: should he/she analyze satellite remote sensing data about rain-forest depletion? or time series of global warming? or decode the human genome? or make sharp, informative real-time images of the beating heart? or make seismic images of the Earth with a view to understanding earthquakes and volcanoes? At some point, an essential singularity is going to occur: statisticians' work time will get completely swallowed up by all the interesting projects, no one will have time to do even minimal PR, and the profession will disappear from public view entirely.

Of course, "disappearing from public view" doesn't mean that our profession has no chance for glory and prestige. My undergraduate thesis adviser, John Tukey, received the Presidential Medal of Science; but I don't think of him as having engaged in PR.

Our relative lack of visibility signifies, to me, that our work doesn't fit in with the pre-packaged, short attention span of the popular and political "culture" of the United States of today. The problems we are working on are a bit too complex to generate good "sound bites" for TV. When I was an undergraduate I found the TV "culture" of American society unsatisfying, and statistics became attractive because of its opposition to the laxity of thought encouraged by the TV "culture."

Our concerns, though they change with the changing demands of science and medicine, are at some level durable. They have too much integrity to align with prevailing fashion. When I was choosing a career, some of my classmates thought that by studying the Law they'd find careers cleaning up governmental problems like Watergate; instead, some of them ended up doing the paperwork behind junk bond offerings. My first job as a statistician involved work in oil exploration; that whole industry collapsed in 1984 happily it turns out that the work I was doing had applications in communications and in extragalactic astronomy as well. Moreover, some of the mathematical questions I started thinking about because of things I had learned in oil exploration prompted me to solve some problems in what is basically pure mathematics: inequalities in Fourier Analysis. Industries can boom and burst, but statistics will always be there, and statistical work will have applications beyond the industry or scientific field it was developed for.

Because statistics is relatively hidden from public view, statisticians feel close to one another in a way I don't observe in other professions. Statisticians really arent competing against one another, because they address the problems posed by other disciplines, and there are so many problems to work on that there's plenty of room for everyone. As a result, statisticians are genuinely friendly and welcomin people. I have friends in France, Israel, China, Australia, Germany, all of whome are statisticians.

I focused here on why you might chose statistics as a career, over, say, Mathematics, Computer Science, or some cognate field. I believe that if you like roles (a)--(d) mentioned in paragraph two of this article, you'll prosper in statistics---in a way you wouldn't prosper in those other fields. I haven't said much about what I do these days for research, but if you visit Sequoia Hall and come talk to me, I'll be happy to chat with you about that.
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Brad Efron

From the time I was a little boy until my senior year in college I wanted to be a mathematician. Then I learned that I really wanted to be a 19th century mathematician, the kind who does a little theory, a lot of computation, and some consulting with real scientists. The field of statistics has allowed me to do all three things, in whatever proportions I desired.

Here is an example of three faces of statistics, done in the early 1940's. The naturalist Corbet had spent two years trapping butterflies in Malaya. At the end of that time he constructed a table to show how many times he had trapped the various butterfly species. (See Table 1.) For example, 118 species were so rare that Corbet had trapped only one specimen of each, 74 species had been trapped twice each, etc.

Table 1: Corbet's data on how often species of butterflies were trapped

Corbet returned to England with his table, and asked R. A. Fisher, the greatest of all statisticians, how many new species he would see if he returned to Malaya for another two years of trapping. This question seems impossible to answer, since it refers to a column of Corbet's table that doesn't exist, the ``0'' column. Fisher provided an interesting answer to the question, which was later improved on by I. J. Good and Turing of Turing machine fame.
The Fisher-Good-Turing answer was this: you can expect to trap

new species in two years of additional trapping.
[If you know something about the Poisson distribution you can derive this formula. Here are some hints.

1. Assume that there are N species of butterflies altogether, and that the i-th one will be trapped a Poisson number of times in two years, with Poisson parameter .
2. Notice that the probability that species i will not be trapped in the first two years, but will be trapped in the second two years, is
   
   

The quantity following the estimate 75 is a ``standard error,'' an estimate of accuracy for the number 75. My main interest in the past several years has been in developing computer-based methods for obtaining standard errors (and other measures of accuracy) in very complicated situations. Classically, quantities are calculated from formulas that grow quickly more intricate and less useful as the estimator of interest gets less like a simple mean. The bootstrap uses the computer to give a numerical value without any formula at all.

Table 2: Data on blood cholesterol decrease

Here is a simple but real example. Nine men who were participating in a medical experiment recorded the following decreases in their blood cholesterol levels. (See Table 2.) These numbers have mean , where the standard error 10.13 is calculated from the time-honored formula

But what if we are interested in an estimate other than the mean, for example the median, equal to 32.50 for this data set. There is no standard error formula for the median. The bootstrap estimates the standard error of the median by repeatedly drawing ``bootstrap samples" from the original data, reevaluating the median for each bootstrap sample, and estimating the standard error of the original median by the observed variability in the bootstrap medians. A bootstrap sample is a sample of the original size, 9 in this case, drawn randomly but with replacement from the original data set. For the cholesterol data, 400 bootstrap samples yielded 400 bootstrap medians, giving a estimate of 13.59. Notice that the median seems to be a worse estimate than the mean in this case, in the sense of having a bigger standard error.

Try this bootstrap calculation yourself!

A great deal of theoretical work, by many statisticians, has gone into showing that the bootstrap algorithm works. In situations where there exists a formula, like the case of the mean, the algorithm produces nearly the same number as the formula. Whether or not a formula exists the algorithm produces a number that has excellent theoretical properties as an accuracy estimate. Finally, and most importantly, the bootstrap's good theoretical properties carry over into real statistical practice.

My current research interests center on computer-based statistical methodology. I am trying to find computer algorithms that automate the often intricate methods of classical statistical inference. The goal is both a more flexible theory, and a better understanding of the classical methods.
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Confessions of a Scientific Dilettante (name withheld on request)

I can remember the difficulty I had choosing an undergraduate major and the absolute horror of selecting a single subject to study in graduate school. Although I still have occasional misgivings about the inevitable degree of specialization demanded of contemporary scientists, I have had very few serious regrets about my choice of a career as an academic statistician. The principal reason is that I can still dabble in a number of scientific subjects at varying degrees of seriousness and spend a substantial part of my time thinking about interesting mathematical problems. My work is largely theoretical, and therefore I spend more effort thinking about mathematics and computing than about applied science. Yet I have spent a great deal of time in recent years pondering the proper formulation of problems of sequential clinical trials and of change-point problems in quality control and epidemiology. I also like to watch from a distance the progress of stochastic models in finance, change-point like problems in genetics, and some stochastic models in physics and physical chemistry. Statistical consulting presents another opportunity for intellectual variety and perhaps financial remuneration, which I used to find necessary each year at income tax time. The variety of problems is enormous, and for each there are different approaches to a solution. The only limits are my own speed at absorbing new ideas (usually not as fast as I would like) and the twenty-four hours in each day.

An area of recurring interest for me is Sequential Analysis. The foundation of this subject was laid by Abraham Wald during World War II, when it was applied by the U.S. Military to problems of sampling inspection. Today the applications motivating the greatest activity are clinical trials involving human subjects, where ethical considerations require careful monitoring of data to insure that information about treatment effectiveness or unfavorable side effects is discovered as soon as is reasonably possible. A subject which is quite different conceptually but involves similar mathematical ideas is Stochastic Control Theory, which may be applied to guidance systems for ships or rockets or to the analysis of investment portfolios.

Of the problems I have worked on seriously, I believe the class of Change-Point problems offers some of the greatest variety and challenges. Originally these problems arose in quality control, where one observes the output of a manufacturing process the quality of which is subject to random variability and can be "in control" or "out of control." The state of the process can not be observed directly, but must be inferred by observations on the quality of the output of the process. A change-point detection scheme is one which observes the output sequentially and occasionally signals that a process is out of control. The procedure must be constrained to signal very rarely when the process is not out of control, and subject to this constraint to detect as quickly as possible a true lack of control. Similar problems involving sequential detection of change-points arise in monitoring public health records for the onset of an epidemic or an increase in the rate of occurence of congenital abnormalities. Change-point problems also arise in retrospective analysis of a variety of processes evolving over time, e.g., in econometric time series. They lead naturally to novel probability problems involving distribution of maxima of random processes and random fields; and they involve interesting questions close to the foundations of statistical inference: questions of ancillarity and of the relations between Bayesian and likelihood methods.

I am also interested in Probability Theory, especially random walk, renewal theory, and Brownian motion, which play an important role in both sequential analysis and change-point problems. I find the probability theory arising in the context of these statistical problems to be particularly interesting, and on numerous occasions have succumbed to the fascination of probability theory without regard to its relation to statistics.
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