Celebratio Mathematica

David H. Blackwell

A Tribute to David Blackwell

by Roger J.-B. Wets

I first thought I would de­vote this short con­tri­bu­tion to a couple of re­mark­able tech­nic­al achieve­ments of Dav­id Black­well and how they in­flu­enced sub­sequent re­search. This would have in­cluded the deep in­sight provided by his el­eg­ant proof of Lya­pun­ov’s the­or­em about the range of a vec­tor meas­ure, about his sem­in­al art­icles lay­ing the found­a­tions of dy­nam­ic pro­gram­ming, and so on. But it is in his role as a lifelong ad­visor and mod­el that his in­flu­ence turned out to be most sig­ni­fic­ant.

It is im­possible to find any in­form­a­tion about Dav­id that does not refer to him as an out­stand­ing teach­er, and in­deed he was. He liked his classes to be sched­uled as early as reas­on­able. The first course I took with him was an un­der­gradu­ate course on dy­nam­ic pro­gram­ming, in which he mostly covered his own de­vel­op­ment of the field. It was lis­ted as an un­der­gradu­ate course, I sup­pose, on the basis that he didn’t re­quire much more than a de­cent back­ground in real ana­lys­is and lin­ear al­gebra. But one could nev­er have guessed that it was an un­der­gradu­ate class on the basis of the stu­dent body. There might have been one or two smart un­der­gradu­ates lost in the audi­ence, but the rest con­sisted mostly of gradu­ate stu­dents in op­er­a­tions re­search and stat­ist­ics and a not in­sig­ni­fic­ant num­ber of fac­ulty mem­bers. In ad­di­tion to re­mem­ber­ing that home­work as­sign­ments were ex­tens­ive, in­struct­ive, and re­l­at­ively hard, I was fas­cin­ated by the con­struct­ive ap­proach; not just wheth­er it ex­ists or might be done but the fact that the res­ults were de­rived in such a way that sug­ges­ted the po­ten­tial of solu­tion pro­ced­ures. I didn’t real­ize at the time how strongly it would even­tu­ally in­flu­ence my own re­search strategy.

After I took a couple more courses with him and chose to work in stochast­ic op­tim­iz­a­tion, Dav­id be­came a nat­ur­al coad­visor of my thes­is. The sub­ject stochast­ic pro­gram­ming (de­cision mak­ing un­der un­cer­tainty) had been pro­posed by G. B. Dantzig. I was pleased but not sur­prised by Dav­id’s ac­cept­ance to act as coad­visor. But his ad­vice/com­ments could be quite can­did and to the point. The first time I went to dis­cuss what I was plan­ning to do, I gave a too-suc­cinct ver­sion of the class of ques­tions I was go­ing to con­sider, and Dav­id bluntly told me “but that’s just find­ing the min­im­um of an ex­pec­ted func­tion”, and he def­in­itely was not im­pressed. When, a bit later, I ex­plained that this “func­tion” was not a simple one but in­volved not just an ob­ject­ive but also (com­plex) con­straints, he re­vised his as­sess­ment to “Oh, that, make sure you first handle some man­age­able cases”, and he im­me­di­ately star­ted with a couple of sug­ges­tions that even­tu­ally turned up as il­lus­tra­tions in my thes­is.

He had played, more than once, the role of the “wise uncle” for stu­dents in­ter­ested in op­tim­iz­a­tion who were con­cerned about get­ting a de­gree in a field whose math­em­at­ic­al stand­ing wasn’t yet well es­tab­lished or re­cog­nized. They some­how felt that they could con­fide their con­cerns to him and would then re­ceive the ap­pro­pri­ate ad­vice. He could be quite plain­spoken in such situ­ations and simply told the hes­it­at­ing stu­dent, “You are telling me that you are in­ter­ested in area A, but would con­sider get­ting a de­gree in stat­ist­ics, how can this make sense?” For one of my friends, this ad­vice turned out to be ex­actly what was needed, and it res­ul­ted in a bril­liant, math­em­at­ic­ally rich ca­reer.

I didn’t re­turn to stat­ist­ics un­til it be­came dif­fi­cult to ig­nore the ubi­quit­ous lack of stat­ist­ic­al data avail­able to con­struct re­li­ably the dis­tri­bu­tion of the ran­dom quant­it­ies of a stochast­ic op­tim­iz­a­tion prob­lem. My ap­proach was based on the idea of in­cor­por­at­ing in the es­tim­a­tion prob­lem all the in­form­a­tion avail­able about the stochast­ic phe­nom­ena, not just the ob­served data but also all nondata in­form­a­tion that might be avail­able, and re­ly­ing on vari­ation­al ana­lys­is for the the­or­et­ic­al found­a­tions and op­tim­iz­a­tion tech­niques to de­rive non­para­met­ric, as well as para­met­ric, es­tim­ates. This didn’t look like an easy sale to either fre­quent­ist or Bayesian stat­ist­i­cians. So, I went to see Dav­id, by then pro­fess­or emer­it­us. After all, this could be fit­ted in the frame­work of the the­ory of games and stat­ist­ic­al de­cisions. This time, it didn’t take him more than a few minutes to un­der­stand and en­cour­age me to pur­sue this ap­proach. Of course, he also im­me­di­ately sug­ges­ted fur­ther pos­sib­il­it­ies and re­served a place for a lec­ture in the Ney­man Sem­in­ar, as well as time for fur­ther dis­cus­sions.

On re­peated oc­ca­sions, Dav­id provided this steady an­chor that made you feel that what you were try­ing to do was or was not worth­while, and, giv­en the wide scope of his in­terests and know­ledge, this al­ways turned out to be an in­valu­able re­source. Thanks, pro­fess­or ex­traordin­aire, Dav­id Black­well.