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Fragment of statistical autobiography

I studied math and physics at MIT. To be more precise, I started in math as default–ever since I was two years old, I’ve thought of myself as a mathematician, and I always did well in math class, so it seemed like a natural fit.

But I was concerned. In high school I’d been in the U.S. Mathematical Olympiad training program, and there I’d met kids who were clearly much much better at math than I was. In retrospect, I don’t think I was as bad as I’d thought at the time: there were 24 kids in the program, and I was probably around #20, if that, but I think a lot of the other kids had more practice working on “math olympiad”-type problems. Maybe I was really something like the tenth-best in the group.

Tenth-best or twentieth-best, whatever it was, I reached a crisis of confidence around my sophomore or junior year in college. At MIT, I started right off taking advanced math classes, and somewhere along the way I realized I wasn’t seeing the big picture. I was able to do the homework problems and do fine on the exams, but something was missing. Ultimately I decided the problem was that, in the world of theoretical math, there were the Cauchys, the Riemanns, etc., and there were everybody else. I didn’t want to be one of the “everbody else.” Unfortunately I didn’t know about applied math at the time–at MIT, as elsewhere, I imagine, the best math students did the theory track.

I was also majoring in physics, which struck me as much more important than math, but which I felt I had even less of an understanding of. I did well in my classes–it was MIT, I didn’t have a lot of friends and I didn’t go on dates, so that gave me lots of time to do my problem sets each week–and reached the stage of applying to physics grad schools. In fact it was only at the very last second in April of my senior year that I decided to go for a Ph.D. in statistics rather than physics.

I had some good experiences in physics, most notably taking the famous Introduction to Design course at MIT–actually, that was a required course in the mechanical engineering department but many physics students took it too–and working for two summers doing solid-state physics research at Bell Labs. We were working on zone-melt recrystallization of silicon and, just as a byproduct of our research, discovered a new result (or, at least it was new to us) that solid silicon could superheat to something like 20 degrees (I think it was that, I don’t remember the details) above its melting point before actually melting. This wouldn’t normally happen, but we had a set-up in which the silicon wafer was heated in such a way that the center got hotter than the edges, and at the center there were no defects in the crystal pattern for the melting process to easily start. So it had to get really hot for it to start to melt.

Figuring this out wasn’t so easy–it’s not like we had a thermometer in the inside of our wafer. (If we did, the crystalline structure wouldn’t have been pure, and there wouldn’t have been any superheating.) We knew the positions and energies of our heat sources, and we had radiation thermometers to measure the exterior temperature from various positions, we knew the geometry of the silicon wafer (which was encased in silicon dioxide), and we could observe the width of the molten zone.

So what did we do? What did I do, actually? I set up a finite-element model on the computer and played around with its parameters until I matched the observations, then looked inside to see what our model said was the temperature at the hottest part of the wafer. Statistical inference, really, although I didn’t know it at the time. When I came to Bell Labs for my second summer, I told my boss that I’d decided to go to grad school in statistics. He was disappointed and said that this was beneath me, that statistics was a step down from physics. I think he was right (about statistics being simpler than physics), but I really wasn’t a natural physicist, and I think statistics was the right field for me.

Why did I study statistics? I’ve been trained not to try to answer Why questions but rather to focus on potential interventions. The intervention that happened to me was that I took a data analysis course from Don Rubin when I was a senior in college. MIT had very few statistics classes. I’d taken one of them and liked it, and when I went to a math professor to ask what to take next, he suggested I go over to Harvard and see what they had to offer.

I sat in on two classes: one was deadly dull and the other was Rubin’s, which was exciting from Day 1. The course just sparkled with open problems, and the quality of the ten or so students in the class was amazing. I remember spending many hours laboriously working out every homework problem using the Neyman-Pearson theory we’d been taught in my theoretical statistics course. It’s only by really taking this stuff seriously that I realized how hopeless it all is. When, two years later, I took a class on Bayesian statistics from John Carlin, I was certainly ready to move to a model-based philosophy.

Anyway, to answer the question posed at the beginning of the paragraph, Don’s course was great. I was worried that statistics was just too easy to be interesting, but Don assured me that, no, the field has many open problems and that I’d be free to work on them. As indeed I have.

Why did I start a blog? I realize I’m skipping a few steps here, considering that I started my Ph.D. studies in 1986 and didn’t start blogging until nearly two decades later. I started my casual internet reading with Slate and Salon and at some point had followed some links and been reading some blogs. In late 2004 my students, postdocs, and I decided to set up a blog and a wiki to improve communication in our group and to reach out to others. The idea was that we would pass documents around on the wiki and post our thoughts on each others’ ideas on the blog.

I figured we’d never run out of material because, if we ever needed to, I could always post links and abstracts of my old papers. (I expect I’m far from unique among researchers in having a fondness for many of my long-forgotten publications.)

What happened? For one thing, after a couple months, the blog and wiki got hacked (apparently by some foreign student with no connection to statistics who had some time on his hands). Our system manager told us the wiki wasn’t safe so we abandoned it and switched account names for the blog. Meanwhile, I’d been doing most of the blog posting. For awhile, I’d assign my students and postdocs to post while I was on vacation, but then I heard they were spending hours and hours on each entry so I decided to make it optional, which means that most of my cobloggers rarely post on the blog. Which is too bad but I guess is understandable.

Probably the #1 thing I get from posting on the blog is an opportunity to set down my ideas in a semi-permanent form. Ideas in my head aren’t as good as the same ideas on paper (or on the screen). To put it another way, the process of writing forces me to make hard choices and clarify my thoughts. The weakness of my blogging is that it’s all in words, not in symbols, so quite possibly the time I spend blogging distracts me from thinking more deeply on mathematical and computational issues. On the other hand, sometimes blogging has motivated me to do some data analyses which have motivated me to do new statistical research.

There’s a lot more that I could say about my blogging experiences, but really it all fits in a continuum with the writing of books and articles, meetings with colleagues, and all stages of teaching (from preparation of materials to meetings with students). One thing that blogging has in common with book-writing and article-writing is that I don’t really know who my audience is. I can tell you, though, that the different blogs have much different sets of readers. My main blog has an excellent group of commenters who often point out things of which I’d been unaware. At the other blogs where I post, the commenters often don’t always understand where I’m coming from, and all I can really do is get my ideas out there and let people use them how they may. In that way it’s similar to the frustrating experience of writing for journals and realizing that sometimes I just can’t get my message across. In my own blog I can go back and continue modifying my ideas in the light of audience feedback. My model is George Orwell, who wrote on the same (but not identical) topics over and over again, trying to get things just right. (I know that citing Orwell is a notorious sign of grandiosity in an author, but in my defense all I’m saying is that Orwell is my model, not that I have a hope of reaching that level.)

31 Comments

  1. Manoel Galdino says:

    Right now I'm starting to write all that stuff to apply to postdoc, and it includes, of course, some autobiographical information.

    It's very nice to hear from you how did you make your choices and why. It helps me in many ways you probably dont ever imagine.

    Thanks
    Manoel

  2. Koala says:

    Interesting — Ed Witten's story is similar to yours, in that, everyone needs to find that one thing that they are best at.

    http://icio.us/02hhqk

    Also: an observation I have is that everyone seems to have an envy of some kind — economists have physics envy, CS people have math envy, and I've seen some chemists, physicst who had an IT envy.

    Is theoretical math the highest (purest) form of analytical thought? I am not sure. I know many anectodes to support either side. I also know humans evolved according to their technology / tools [1], so deriving pages of formulas in one sitting maybe is not in our evolutionary future. I dont know. But I do like theoretical stuff.

    Hmm…

    [1] http://www.newscientist.com/article/mg20727741.70

  3. ceolaf says:

    Can you say more about being trained not to answer why questions, but rather to focus on potential interventions?

    I think this is CRITICAL aspect of the nature and contribution that statistics makes to other fields, one that few people really understand.

  4. Bob Carpenter says:

    I've heard variants of this story from most of the people I know who work in mathematically oriented research outside of math or physics itself. Like myself.

    I wanted to do theoretical computer science, specifically logic and computability theory, not physics, but the issue's the same. Around the second year of college, where I was a pure math major, I started taking upper-level math classes (like analysis and abstract algebra), and I realized the really good math majors were orders of magnitude better than I was at problem solving.

    You see the same thing in programming. If you go to sites like TopCoder, you'll see the winners of the contests doing amazing amounts of tight programming in minutes. Sort of like the Putnam contest, with the difference being I could at least do all the programming in a few hours, whereas the Putnam had problems I couldn't solve in a month or perhaps ever.

    In retrospect, I too have a couple of questions about what's going on. One, how much of this ability is trainable? And when? Two, how much of research involves this kind of narrow problem solving, and how much other factors, such as being able to formulate the right problem?

    Luckily, the real world isn't like a timed closed-book test where you have to work alone.

  5. Numeric says:

    It's interesting that you note that Orwell is your model. There is something Orwellian about Bayesian theory. In particular, it would be interesting for a historian of science to trace when an improper prior started being referred to as a non-informative prior. On the other hand, all of the historical record has probably been flushed down the memory hole.

  6. Anonymous says:

    I've been an avid reader of this blog for some time, and this was one of the most interesting posts I've read.

    I was a physics major in college as well, and I also had a crisis of confidence and went to a related field. Essentially, I realized that I rarely solved problems all the way through correctly on my own. I could "work the system" by going to office hours and talking to other students, and thus my HW grades were excellent and test grades were average. But I felt like there were other students who could independently solve almost all problems correctly entirely on their own, and who wouldn't get stuck on all variety of mathematical issues.

    I certainly discovered that the more of a similar sort of problem I did, I would get better at that narrow class of problems. But in general, I felt like I wasn't necessarily getting any better at any other narrow class of problems.

    So, Dr. Gelman, could you comment on what the role of practice is in solving physics/mathematical problems? Is there some "g" that is actually behind ability to solve new classes of problems with no direct experience, and is that "g" fixed? Or, instead, is this even the wrong question to be asking?

    Also, I too am curious to know what you mean by being trained to avoid Why questions, and instead focusing on interventions…

  7. Jerzy says:

    Bob and Andrew, that sounds familiar. Like many folks I used to believe that this was all about sheer brainpower, and so after a couple of ridiculously hard classes I shifted from pure math to applied things like engineering and statistics.
    But now I'm starting to feel that quite a lot of this hardcore math/science ability is learnable after all. It's more like learning fluency in a language and being able to catch things that you'd otherwise miss, rather than having raw processing power in your head.
    In any case, in practical terms it's far more useful to formulate the right problem than it is to solve narrow problems a bit faster than others.

    Physicist Frank Wilczek puts it better than I can:
    "To anyone who reflects on it, it soon becomes clear that F = ma by itself does not provide an algorithm for constructing the mechanics of the world. The equation is more like a common language, in which different useful insights about the mechanics of the world can be expressed. To put it another way, there is a whole culture involved in the interpretation of the symbols. When we learn mechanics, we have to see lots of worked examples to grasp properly what force really means. It is not just a matter of building up skill by practice; rather, we are imbibing a tacit culture of working assumptions. Failure to appreciate this is what got me in trouble."
    http://xserver2.lns.mit.edu/~csuggs/physics_today

  8. Andrew Gelman says:

    Numeric:

    I recommend weakly informative priors. See some of our recent research articles for examples, in particular our 2006 Bayesian Analysis paper and our 2008 Annals of Applied Statistics paper.

  9. Igor Carron says:

    Andrew,

    I am sure you ahve heard of the Orwell blog ( http://orwelldiaries.wordpress.com/ ).

    Igor.

  10. Andrew Gelman says:

    Igor: Thanks for the link. Scary stuff. But you can hardly blame someone for what's in his diary, right?

  11. Bill Jefferys says:

    I, too, found this very interesting. Like Andrew, I was good at math but interested in a physical science (astronomy) from an early age. At Wesleyan, I had outstanding math and physics professors. Had I taken topology, I would have had a math minor, and had I taken E&M, a physics minor. But I majored in astronomy, got my Ph.D. in that field (in a mathematical area, celestial mechanics) and went on to the University of Texas for 40 years in the astronomy department.

    But, as I say, I was always interested in math, and my dissertation in celestial mechanics was just the beginning. I did postdoc work at NYU with Juergen Moser (of KAM theory fame), which was a wonderful experience. At Texas, I was the designated professor to teach our "Mathematical Methods for Astronomy" course, basically elementary applied statistics of the sort useful to astronomers.

    But teaching that course for many years gave me concerns. I began to wonder about the intellectual ideas behind confidence intervals and hypothesis testing; they were hard to explain to the students and seemed to lack something. But at the time I was inhabiting "sci.math.stat" on UseNet, and I kept reading about this mysterious Bayesian idea. So I started reading about it. About that time, Jim Berger and Don Berry published an article in American Scientist that was critical of classical hypothesis testing, and I contacted Jim, who graciously answered my too-numerous questions and assigned reading. That is what turned me into a Bayesian. I had a very productive sabbatical with Jim at Duke, and have worked with him from time to time since then.

    So, after I retired, and left Texas for Vermont, I offered to teach Bayesian stats at UVM, since they did not have a course. They were happy to offer me an adjunct position, and I have taught the course from time to time since.

  12. Bill Jefferys says:

    Actually, I was missing one course in math and one course in physics to have majored in those subjects. I probably should have taken E&M, because it is important in astronomy (but I got enough in grad school). I never missed topology, and was fortunate to have taken functions of a complex variable, which has been useful to me for most of the past 50 years. It seems that math majors don't get this course any more. That's too bad, if true.

  13. Phil says:

    I don't have much to add, except from the perspective of someone who has known Andrew since 7th grade. Andrew was always exceptional at math — far better than me — and at the sorts of physics and engineering problems and puzzles that came up in junior high and high school. Speaking as someone who majored in physics (with mediocre grades) and went on to get a PhD in theoretical atomic physics (at a mediocre grad school), Andrew certainly could have been a good physicist. Better than me there, too, no doubt.

    But I agree with Andrew that although he would have been a very good physicist, he wouldn't have been brilliant. I detest Ayn Rand and all that she stands for now, but when it comes to physics it does seem to me that her worldview applies pretty well: there are some exceptional people who are capable of making enormous strides, while most of the rest of us only trudge forward a few inches at a time, or, worse, get in the way. Back in 1900 you could have put 100 physicists like me, or 50 mathematicians like Andrew, to work on general relativity, and by the end of our careers 40 years later we still wouldn't have had what one Einstein did in a couple of years. Some people hit 7-7-7 on the one-arm bandit of brainpower, those people are just way ahead of those of us who got 7-7-cherry.

    Physics (or math, or whatever) isn't alone in this. The best chess player you've ever played will make you look like a complete chump, but he'd lose a match to a world-class player even if the world-class player had to play blindfolded and only had thirty seconds per move. And I'm not kidding. A phrase often used in sports to summarize this phenomenon is "levels of the game," but it really applies to all areas of human endeavor. Even among people who self-select for skill in a particular area, the one-in-a-million practitioner is far, far, far better than the 1-in-a-thousand, and the 1-in-a-thousand is far, far better than the 1-in-100, and the 1-in-100 is a lot better than the 1-in-10, and the 1-in-10 is a lot better than average. That's me down there at "average", and Andrew at 1-in-10 or even 1-in-100. If you're in the top 1% among mathematicians or physicists, you are absolutely brilliant compared to the average person, but Isaac Newton could give you a 100-yard head start, possession of the ball, rook and move, choice of end, and use of your favorite cue, and he'd still beat the tar out of you at whatever bizarre game you are playing with all of those implements. (Sorry, my metaphor got out of hand).

    Ideally, each of us finds something we're great at, and that's the field we choose. But most of us aren't truly great at anything, we just have to find something we're tolerably good at. For me, I sort of drifted into physics as something that would let me keep going to school without having to get a real job. Then I got out in the real world and found a comfortable and fascinating niche at the intersection of physics, engineering, and data analysis.

    If there's a moral here, and maybe there is, it's that Andrew made a smart choice at the end of college, when, rather than simply take the path of least resistance, he did some conscious thinking about what field to pursue. Too many people just drift along and end up going into an area that isn't a good fit. Do as I say, don't do as I did: follow Andrew's lead on this. But also realize that all is not lost even if you don't think about it. I've had a happy and productive career so far, with less talent and less forethought than Andrew put into his.

  14. zbicyclist says:

    This post is interesting, as are the comments. It's particularly interesting since I come from practically the opposite background.

  15. William Ockham says:

    Another thing about blogging and commenting is accountability. In that spirit, i hereby acknowledge that my election prediction was terribly wrong.

  16. Steve Sailer says:

    Thanks.

    At a level a standard deviation or two lower in IQ, I had a roughly similar experience: I hit my continuous math ceiling freshman year in college, but then finally got fascinated by statistics my last quarter of my MBA.

    I took from that the lesson that the educational system should have taught me more statistics earlier. But thinking back on it, it could be that I just wasn't mature enough for statistics. I took statistics courses my senior year in college and my first year at MBA school, but they weren't that interesting to me.

    It could be that you have to reach a certain level of maturity to appreciate statistics. When you are young, you hunger for the kind of absolute truth that math and physics provide, but when you are older, you enjoy thinking about probabilistic patterns.

    It's kind of like how writers can change from being lyric poet to social novelists as they age. They are more brilliant when they are young, but they know more stuff about people when they are old.

  17. koala says:

    This maybe little off topic but, the type of math studied could matter a lot in scientfic career selection and Gilbert Strang from MIT complains that the curriculum in most technical schools is too heavy on Calculus.

    http://www.math.temple.edu/~siagla/articles/Stran

    I wonder if Calculus (or other Physics related subfields) is that necessary for today's issues, problems. I mean both in terms of basic research and engineering.

  18. Andrew Gelman says:

    Steve:

    Regarding maturity and statistics, see Dick De Veaux's talk, Math is like music, statistics is like literature.

  19. koala says:

    Steve:

    I have always felt there was something wrong with the way natural science was taught in schools, at least in schools that I went to. The boundary between formulas and real world never got enough attention. Maybe due to this frustration people later in life take a look at statistics and go "that's what was missing!".

    Two more things on this topic:

    I routinely see great mathematicians who make forays into probability, statistics. The famous Dantzig story is a great example — the one where he comes late to class, sees two problems on the board, etc.. That one. This fits with De Veaux's talk above too; this is how we think rationally. This is the science of knowing what we dont know, so naturally mathematicians are active in this area.

    On IQ: Malcolm Gladwell makes some good points on IQ on one his books; According to him IQ, as a sign of raw processing power, beyond certain level does not make a difference. At this point, more imagination is necessary. Einstein has a similar saying "imagination is more important than knowledge".

  20. K? O'Rourke says:

    Many aspects here – a few comments.

    Maybe there is an economy of scale to posting blogs but the few I have done have not been of adequate cost/benefit to me – confounded of course with topic and poster but very little feed back – but there was always “the post staring back at me”.

    As for math background, I had very little and remember telling my biostats adviser that I felt I really needed to learn some linear algebra. He said “just buy a Schaums outline and learn it this weekend”. Instead I enrolled in the year long math major’s 1st linear algebra course and out of a class of 80 I got about the 5th highest mark, but most others were 1st year students and I was a graduate student. (And there was one “jerk” who got 100% on almost every test and assignment.) But it was probably the best thing I did – training helps and often there is an initial very steep learning curve to get over. But in the end its – “survival of the fit” not the “fittest”. You end up fitting in somewhere where you can contribute. Unfortunately in the usual regression to the mean sense – only the more successful tend to talk about where the ended up always getting the why wrong.

    I am not sure how important math per se will be in statistics in the future – abstract thinking yes, but calculus type manipulations and analysis type rigour – I doubt it. As an aside the Canadian statistical community has been thrown in with mathematicians for most funding opportunities – perhaps it seemed as if they were just trying to do hard math so the funding agencies thought this would be best for them!

    In a Peircian close, he identified three aspects – Whys (explanations) which are always wrong, Musts (implications) for which we have no excuse getting wrong (no mistakes in math) and Shoulds (inductions) for which with randomization we get upper bounds on how often we will be wrong. Not sure if this was what Andrew was referring to.

    K?

  21. koala says:

    Third point: I remember Dr. Gelman write once: "Math is smarter than I am" — math is a technology. It is not simply a collection of symbols for showing off how smart we are (that's what I understand from it anyway). It let us do things we would not be able to do otherwise.

    Another anectode is from Julius Schwinger / Feynman rivalry. Feynman diagrams replaced the derivations of Schwinger and he wasnt happy about that. He called Feynman diagrams "pedagogical improvement". But isnt that improvement enough? Simplifying is a good thing IMHO. And it isnt always easy either.

  22. Paul says:

    The concerns about being eclipsed by giants in a field reminded of advice I read once, I think by Scott Adams (of Dilbert fame). Essentially, a rounding error away from none of us will be at the top of a field. But if you're good at two or three things, combinatorics puts you in a very small group. If you're good at theoretical math and good at explaining concepts in writing, you could be the best mathematics author of a generation. Or if you master geology, statistics and biology, maybe there's unique contributions you can provide from that particular vantage point.

  23. K? O'Rourke says:

    koala: Thanks for the Strang pdf – I need to talk to my daughter about a course after her 1st year calculus course and this will be helpful – her current target is environmental law – and we all know she will need some understanding of stats ;-)

    I'll have to think what it is now about linear algebra thats helpful for stats – certainly the abstract representations but maybe linear structures are too constrictive now. (It would help some spot problems with Correspondence Analysis programs as was the case with SAS implementation of it)

    My guess is complex analysis (at some level) is the real target being a nice mix of calculus and linear algebra… and as Tukey once told me the most appropriate way to think about numerical integration (on the circle)

    K?

  24. Jenny says:

    It is a fascinating post, and the comments are very interesting to me also. I opted out of math and physics at a much earlier stage – my AP physics class was probably my favorite class senior year of high school, and I was always effortlessly the best at math in my (very small) HS class, but I got to Harvard and took a look around freshman year and saw that truly it was just a minor activity of interest and enjoyment for me and a passion for others! I am certain that if I were in a scientific field it would be something like epidemiology or computer science, with a strong applied component – but as I am more theoretically than practically inclined, it is perhaps just as well that I am an English professor. Certainly reading and writing are (to an extent I had no perspective on when I was younger, they seemed so natural to me) my true lifelong obsession – I was writing constantly and reading probably 400+ books a year from age 5 to 18, so it is hardly surprising that my talents in this realm should so much exceed the other things I liked as a teenager and had some modest talent for (physics, music) but didn't pursue obsessively.

  25. Bob Carpenter says:

    @Paul: Exactly! We were just talking about this with Andrew before our meeting this morning. It's all about the bag of skills and how they work together.

    Math is to science as strength, speed and endurance are to sports. It almost always helps to be faster and stronger if you're an athlete, and it almost always helps to be better at math if you're a scientist.

    The problem is that being strong and fast isn't enough for a sport like basketball. In the same way, being great at math isn't enough to make one good at stats or computer science.

    A basketball player has the choice of working on his or her free throw or dribbling, or hitting the weights or going running. Both kinds of activity help, but which one helps most depends on the player's distribution of skills.

    One of the reasons math comes up in these contexts is that skill in math is (relatively) easy to measure.

  26. K? O'Rourke says:

    Bob: I like your analogy but would spin it to "math for working out the right Musts (implications), creativity for comming up with novel Whys (explanations) and wisdom for getting the less wrong Shoulds (inductions).

    But a lot of the Musts now can be rightly worked out with simulations (OK just pointwise) so math may be getting less important or at least its not as big a barrier as it used to be.

    K?

  27. Nice post. I have followed this blog for at leats 2 years, and it is one of my prefered source of ideas about research and teaching.
    Although I was good at math during high school, I decided to study Medicine, instead of engineer as my advisor suggested me then. In my first year, I studied Biostatistics and randomness appeared to me as an important explanation of many phenomena in physiology, histology, … However, most of my professors taught me these courses with a deterministic point of view, till I took a course on Epidemiology. After that course, I felt that practicing medical care was not my personal aim, and decided to train me in statistical methods.
    Nowadays I teach biostatistics in a medical school in Seville, Spain. My main challenge is to convince my students that in most medical activities, the physician is making a decision under uncertainty. It is difficult to change their minds, because their are prone to see the decision as a one-way process.
    Surfing the web several year ago, I found a title, "Teaching statistics: a bag of tricks". This book opened my mind to new ways for teaching my courses on biostatistics. The key: the students learn by doing, not by teaching the teacher.
    The next step was I discovered that one of the author posted in a blog. Blogging was out of my scope, and I didn't consider it a teaching tool. However, after reading the posts every day, I started to think that a blog can become an engaging way to discuss with my students. Thanks, Andrew.

  28. koala says:

    K?: I need to read up on complex analysis, thanks for the pointer. My path into bayesian stats was through chaos theory, linear dynamic systems, machine learning; I did a lot of algebra with multivariate distributions, hmm, kalman filters, lda, pca; almost everything I dealt with had a linear algebra component in it. After Dr. Gelman mentioned it I also remembered FEM, there is a great section on it in one of Strang's books; I will have to look into at some point.

  29. JSE says:

    "Ultimately I decided the problem was that, in the world of theoretical math, there were the Cauchys, the Riemanns, etc., and there were everybody else. I didn't want to be one of the "everbody else.""

    That's your right, of course, and obviously your choice of career has worked out splendidly for you. But you know who else isn't as good as Cauchy, or Riemann, or Serre, or Grothendieck? Me, and all your colleagues at Columbia, and just about every mathematician you've ever met. We lose tons of students, who would make very good mathematicians, but who feel that somebody else is smarter, faster, more technically powerful than they are. It's true, but so what? If math were done only by the ten best mathematicians in the world, math would die.

    "[in physics] there are some exceptional people who are capable of making enormous strides, while most of the rest of us only trudge forward a few inches at a time, or, worse, get in the way."

    I can't speak to physics, but this is certainly not the case in pure math, unless the class of "exceptional" people numbers in the thousands, and includes lots of people who did not sail effortlessly through ever math course in college.

  30. Andrew Gelman says:

    Jse:

    Good point. There's lots of room in math for non-Cauchys. There's room for applied mathematicians, for theorists who work in new and unusual areas, for teachers, for popularizers, and so forth. Somehow this wasn't made clear to me when I was a math student. Instead it was just theorem after theorem. Even the more more conceptual courses (such as topology) and the courses such as complex analysis that were more focused on problem-solving–even they were presented as edifices.

  31. Fuck being the best at something (although I suppose it worked for you -I pay a lot more attention to you than I do to a 2nd tier physicist or theoretical mathematician).

    However, I'm conflicted because I do think theoretical physics and some types of math should be lower status than quantitative methods social science (the direction of optimization capacity should influence the status of fields, it seems to me, and since all fields are social science enterprises, organizational optimization should have something close to status primacy).

    I read this a couple years ago and re-read it today looking for something else. I think you have something of a superhero origins feel to you, given that you're archetypal for your field, and your rare expertise "rescues" us from villians, human and nonhuman alike.