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A Tribute to David Blackwell George G A Tribute to David Blackwell George G. Roussas, Coordinating Editor n July 8, 2010, Professor David Black- Manish Bhattacharjee well of the University of California, Berkeley, passed away at the age of In David Blackwell, we have that extraordinary ninety-one. combination of exceptional scholarship and su- OIn appreciation of this great mathematical perb teaching that all academicians aspire to but scientist, the editor of Notices of the American rarely achieve. The teaching aspect was manifest at Mathematical Society, Steven Krantz, decided to all levels, ranging from a basic introductory course compile a rather extensive article consisting pri- to cutting-edge research. Anyone who has been at marily of short contributions by a number of Berkeley for any length of time is familiar with selected invitees. Krantz, being a mathematician, the fact that his section of “Stat-2” class routinely felt the need for a bridge toward the community had more students than the combined enrollment of statisticians and probabilists. I had the good of all other sections of the same course, which fortune to be invited to fill this role and to be given were typically taught by others. My own intro- the opportunity to offer a token of my affection duction to his teaching style was also through an and immense respect for Professor Blackwell. undergraduate course in dynamic programming A limited number of invitations were sent to that I took in my second or third semester as a a selected segment of statisticians/probabilists. graduate student. His engaging style of explaining The response was prompt and enthusiastic. At the a problem and bringing it to life in this class final stage, twenty contributions were collected I believe, on reflection, was a decisive influence of average length of about one and a half pages. on my choosing stochastic dynamic programming The contributors were selected to represent four and its applications to probability as the primary groups of people: former students of Professor area for my dissertation research. One of my fond Blackwell; former students at the University of memories of how he brought a problem to life in class concerns a story of how, over a period California, Berkeley, but not Professor Blackwell’s of time, he and Samuel Karlin wrestled with the students; faculty of the University of California, problem of proving the optimality of the so-called Berkeley; faculty from other institutions. “bold play” strategy in a subfair “red-and-black” The heart of the present article is the set of game that is an idealized version of roulette. contributions referred to above. The New York Times obituary of July 17, 2010, Deep gratitude is due to the twenty contributors describes him as “a free-ranging problem solver”, to this article for their response to the extended which of course he truly was. His was a mind invitations; for their sharing with the mathematical constantly in search of new challenges, breaking sciences community at large their experiences with new ground in different areas every few years Professor Blackwell; their reminiscences about with astounding regularity. As Thomas Ferguson him; and their expressed appreciation of Professor told UC Berkeley News in an interview recently, Blackwell’s work. Blackwell “went from one area to another, and he’d —George Roussas write a fundamental paper in each.” His seminal George G. Roussas is Distinguished Professor of Statistics Manish Bhattacharjee is professor of mathematical sci- at the University of California, Davis. His email address is ences at the New Jersey Institute of Technology. His email [email protected]. address is [email protected]. 912 Notices of the AMS Volume 58, Number 7 contributions to the areas of statistical inference, countably infinite sequence of lights. A computer games and decisions, information theory, and program is a function f that turns off some lights. stochastic dynamic programming are well known Which lights are extinguished is a function of the and widely acknowledged. He laid the abstract current pattern of on and off lights. You start with foundations of a theory of stochastic dynamic some initial pattern of lights, x0 say, and let the programming that, among other things, led Ralph program run by computing x1 = f (x0), x2 = f (x1), Strauch, in his 1965 doctoral dissertation under and so on inductively, Blackwell’s guidance, to provide the first proof generating the sequence of Richard Bellman’s principle of optimality, which xn; n = 0, 1, 2,.... The se- had remained until then a paradigm and just a quence xn decreases so principle without a proof! it has a limit, which we Each of us who have been fortunate enough to call xω. But for a general count ourselves to be among his students will no function f the program doubt have our personal recollections of him that might not be finished— we will fondly cherish. A recurring theme among xω need not be a fixed such recollections and the lasting impressions they point of f . So continue have left on us individually, I believe, would be his applying f inductively, mentoring philosophy and style. He encouraged once for each countable his students to be independent and did not at all ordinal α arriving at xω1 , mind even if you did not see him for extended the limit at ω1, the first periods as a doctoral student under his charge. He uncountable ordinal. The trusted that you were trying your level best to work limit must actually be things out yourself and waited until you were ready reached at some count- Photo from Paul R.Collection, Halmos di_07305, Photograph Dolph BriscoeAmerican Center History, for U. of Texas at Austin. to ask for his counsel and opinion. A consequence able ordinal, which may David Blackwell early in his of this, exceptions notwithstanding, was perhaps depend on x0. career. that his students took a little longer than average The iterates xα are, of to complete their dissertations (although I have no course, iterates of the function f applied to x0, so hard data on this), but it made them more likely that we may think of the functions fα converging to learn how to think for themselves. (pointwise) to the limit function fω1 , and we may wonder what sorts of functions g have the form of Richard Lockhart such limits if the basic function f is Borel? David looked at the output program fω1 as a step in I was David Blackwell’s Ph.D. student from mid- defining general functions g between two polish 1977 to early 1979, having asked him to supervise spaces U and V , which were defined by mapping U after hearing the clearest lectures I had, and have, into the computer’s state space by a Borel function, ever heard. I was Ph.D. student number sixty- running the program, and then mapping the result two of sixty-five, according to the Mathematics into V in a Borel way. Any function admitting such Genealogy Project, where it will be seen that David a factorization is Borel programmable (BP), and had four students finish in 1978, two in 1979, and sets are BP if their indicator functions are. David’s two more in 1981. I remember the chair outside 1978 paper, establishing the basic properties of his office where I would wait for my weekly half the sets and functions and showing that they had hour. I remember that he suggested a problem to potential as useful objects in probability theory, me by giving me a paper and saying he didn’t think is typical of his work—very clear, very concise, he had done everything there that he could have. questions not answered set forth, and not quite A few months later I gave up and asked to work on four pages long. something he was doing currently—programmable In my thesis I solved one or two minor prob- sets. lems connected with these ideas. I was trying to David gave a talk at Stanford in the Berkeley- clarify the relation between BP-sets, R-sets studied Stanford colloquium series in which he described by Kolmogorov and others, and C-sets. I wanted these objects. I emerged from the talk amazed by to know, for instance, if letting the program, en- how clear and easy it all was. Trying to tell others coding, and decoding functions be BP, rather than about it, I saw quickly that the lecture was a very just Borel, gave you a larger class of sets than the clear presentation of something not so easy at all BP class. (John Burgess did the problem properly and that David had the capacity to organize argu- in Fundamenta Mathematica in 1981 using results ments far, far betterthan I. The idea is simple as he from logic concerning monotone operators, the described it. Imagine a computer represented as a lightface and boldface set hierarchies and game quantifiers.) I cannot remember how I came to re- Richard Lockhart is professor in the Department of Statis- alize that I needed to think not about sigma fields tics and Actuarial Science at Simon Fraser University. His but about collections of sets like the analytic sets email address is [email protected]. which had fewer closure properties. I now think, August 2011 Notices of the AMS 913 though, that David planted the seed of this idea To say he was a quick study is a remarkable in my head one day—just by emphasizing the understatement. When you met with him, he would importance of closure properties to me; I rather look over what you had done, ask a few probing suspect that my thesis would not have taken him questions, suggest an approach or two to your more than a few weeks to put together and that current problem, and send you on your way.
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