Celebratio Mathematica

David H. Blackwell

A Tribute to David Blackwell

by Thomas S. Ferguson

It was my good for­tune to have been a gradu­ate stu­dent in stat­ist­ics at U. C. Berke­ley when Dav­id Black­well joined the fac­ulty there in 1954. The dis­tin­guished stat­ist­i­cians who were there already — Ney­man, Lehmann, Le Cam, Scheffé, Loève, and oth­ers — con­sti­tuted the most ap­proach­able fac­ulty I’ve seen any­where. We stu­dents shared cof­fee and con­ver­sa­tion with them in the af­ter­noons. When Black­well joined the group, he fit right in with his warm hu­mor, his win­ning smile, his mod­esty and his con­geni­al­ity with the stu­dents.

He had an out­stand­ing math­em­at­ic­al repu­ta­tion by that time, hav­ing been in­vited to give an ad­dress in prob­ab­il­ity at the ICM meet­ings in Am­s­ter­dam in 1954. In 1955 he was elec­ted pres­id­ent of the In­sti­tute of Math­em­at­ic­al Stat­ist­ics. Im­port­ant for me per­son­ally was his book with M. A. Gir­shick, The­ory of Games and Stat­ist­ic­al De­cisions, which came out in 1954. At that time, I was work­ing on my thes­is un­der the dir­ec­tion of Lu­cien Le Cam. I took a course from Black­well and read his book, which views stat­ist­ics as a sub­set of the art of mak­ing de­cisions un­der un­cer­tainty. The beauty of this view in­flu­enced me to such an ex­tent that my sub­sequent work did not go so much in the dir­ec­tion of the top­ics of my thes­is but more in the dir­ec­tion of the areas that in­ter­ested Black­well — game the­ory, prob­ab­il­ity, and se­quen­tial de­cisions.

Dave was one of the early Bayesian stat­ist­i­cians, that is, he con­sidered stat­ist­ics, and life as well, as a pro­cess of ob­ser­va­tion, ex­per­i­ment, in­form­a­tion gath­er­ing, and, based on one’s pri­or be­liefs and the out­comes of the ob­ser­va­tions, modi­fy­ing one’s opin­ions and act­ing ac­cord­ingly. Al­though his views cer­tainly in­flu­enced me, I was nev­er a com­plete Bayesian — no stu­dent of Le Cam could be — but of all the Bayesians I know, he was the most per­suas­ive. It was char­ac­ter­ist­ic of him to spread his in­terests over sev­er­al areas rather than to spe­cial­ize in one. It is amaz­ing how he man­aged to pro­duce deep and ori­gin­al res­ults in sev­er­al fields. The un­der­ly­ing theme of his work springs from his Bayesian per­spect­ive: prob­ab­il­ist­ic, se­quen­tial de­cision mak­ing and op­tim­iz­a­tion.

Let me men­tion just a few of his achieve­ments. In prob­ab­il­ity, there is a ba­sic re­new­al the­or­em that goes by his name. There is his work in Markov de­cision pro­cesses in which he con­ceived the con­cepts of pos­it­ive and neg­at­ive dy­nam­ic pro­grams and in which the no­tion of Black­well op­tim­al­ity plays an im­port­ant role.

In stat­ist­ics, there is the fam­ous Rao–Black­well the­or­em and its as­so­ci­ation with a simple meth­od of im­prov­ing es­tim­ates now called Rao–Black­well­iz­a­tion. There is a fun­da­ment­al pa­per of Ar­row, Black­well, and Gir­shick that helped lay the found­a­tion for Bayesian se­quen­tial ana­lys­is. The sub­ject of com­par­is­on of ex­per­i­ments was in­tro­duced by Black­well and Stein in 1952. The no­tion of mer­ging of opin­ions with in­creas­ing in­form­a­tion was in­tro­duced by Black­well and Du­bins in 1962.

In game the­ory, he has ini­tial­ized sev­er­al areas: games of tim­ing, start­ing with Rand re­ports on duels; games of at­tri­tion; the vec­tor-val­ued min­im­ax the­or­em, lead­ing to the no­tions of ap­proach­ab­il­ity and ex­clud­ab­il­ity, etc. He has had a long in­terest in set the­ory and ana­lyt­ic sets. This led to his study of con­di­tions un­der which cer­tain in­fin­itely long games of im­per­fect in­form­a­tion have val­ues. This has had a deep im­pact in the field of lo­gic; lo­gi­cians now call such games Black­well games.

My own pro­fes­sion­al in­ter­ac­tion with him came in 1967–1968. He sug­ges­ted work­ing on a prob­lem in the area of stochast­ic games. In 1958 Gil­lette had giv­en an ex­ample of a stochast­ic game that did not have a value un­der lim­it­ing av­er­age pay­off if the play­ers are re­stric­ted to us­ing sta­tion­ary strategies. Dave called this ex­ample game the “Big Match”. He wondered if the game had a value if all strategies were al­lowed. After work­ing on the prob­lem to­geth­er for a while, we sim­ul­tan­eously and in­de­pend­ently came up with dif­fer­ent proofs of the ex­ist­ence of a value. To me, it was just an in­ter­est­ing prob­lem. But Dave some­how knew that the prob­lem was im­port­ant. It was the first step in show­ing that all stochast­ic games un­der lim­it­ing av­er­age pay­off have a value. This took an­oth­er four­teen years, with many schol­ars con­trib­ut­ing par­tial res­ults be­fore the res­ult was fi­nally com­pletely proved.

Dave Black­well is one of my role mod­els. He in­flu­enced me in my pro­fes­sion­al work and in my per­son­al life. He was a great teach­er, both in the classroom and in con­ver­sa­tions on gen­er­al sub­jects. He had a way of cut­ting through massive de­tail to get to the heart of a prob­lem. He had over sixty Ph.D. stu­dents. But if you count people like me, he had many more stu­dents. His spir­it and his works are still alive in all of us.