Celebratio Mathematica

David H. Blackwell

A Tribute to David Blackwell

by Albert Lo

Dav­id Black­well wrote one of the first com­pre­hens­ive treat­ments of Bayesian stat­ist­ics, and his in­sist­ence on the Bayesian ap­proach is le­gendary. I ap­proached Black­well for a Ph.D. thes­is top­ic in the mid-1970s. He told me to look in the An­nals of Stat­ist­ics, find a top­ic I liked, and come back. After some search­ing, I re­por­ted that ker­nel dens­ity es­tim­a­tion in­ter­ested me. Black­well, star­ing at me with his pier­cing big eyes, said “The top­ic is fine, yet it must be done the Bayesian way.” This was ex­actly what he said to me. Later, when I presen­ted to him a res­ult on the con­sist­ency of the pos­teri­or dis­tri­bu­tion of a loc­a­tion para­met­er with re­spect to a Le­besgue pri­or, he con­cluded mat­ter-of-factly, “It is good since it is al­most Bayesian.” Again, these were his ex­act words. On an­oth­er oc­ca­sion he stated that all the non-Bayesian pa­pers in the An­nals have to be re­writ­ten us­ing a Bayesian ap­proach, and I my­self found this “Bayesian­iz­a­tion” a good source of re­search top­ics.

Black­well al­ways in­sisted on the ex­act­ness and clar­ity of solu­tion. For all his un­doubted math­em­at­ic­al abil­ity, his pref­er­ence was for sim­pli­city over math­em­at­ic­al ab­strac­tion. On the dens­ity es­tim­a­tion prob­lem, he sug­ges­ted mod­el­ing a dens­ity by a loc­a­tion mix­ture of uni­form ker­nels and put­ting a Di­rich­let pro­cess pri­or on the mix­ing dis­tri­bu­tion. The prob­lem was hard then, and after a year and a half of fu­tile search­ing, I had to present an al­tern­at­ive, yet more stand­ard, ap­proach based on ex­pand­ing the square root of the dens­ity in an or­tho­gon­al series with a pri­or on the in­fin­ite se­quence of coef­fi­cients that lies on the shell of a Hil­bert sphere. Upon hear­ing the pro­posed ap­proach, Black­well simply com­men­ted “Al, you are not ready.” To this day, I can still hear his dev­ast­at­ing voice! His opin­ion about the ma­tur­ity/read­i­ness of stu­dents was per­cept­ive; two years later his Bayesian mix­ture dens­ity prob­lem was re­solved with an ex­pli­cit solu­tion that he had an­ti­cip­ated, pre­sum­ably at a time when the stu­dent was ready.

Though there are sug­ges­tions of a good-natured rivalry between Black­well and some of his fam­ous col­leagues, he was not one for dir­ect con­front­a­tion. He was very quiet about the ra­cial in­justice that he en­dured and over­came, nev­er men­tion­ing the sub­ject in my hear­ing. It brings to mind how he handled me as a stu­dent, who had been ex­pertly trained by Berke­ley fre­quent­ists. His only ad­vice to me on how to learn Bayesian stat­ist­ics was to read Part III of De Groot’s text. Though he made some ex­tremely valu­able sug­ges­tions in his nice and gen­tle­manly way, he nev­er really dis­cussed or showed me how to ap­proach a re­search prob­lem, ex­cept by ex­ample. I had to find my own way by ob­serving him and oth­ers (mostly oth­ers) in the de­part­ment. The dis­crim­in­a­tion Black­well ex­per­i­enced may have giv­en him the philo­sophy that one should also be able to fight his own way up, or per­haps that if one is worthy, one will even­tu­ally be able to make it on one’s own. Or per­haps he un­der­stood that this was the right ap­proach to take with cer­tain stu­dents in­di­vidu­ally.

Dav­id Black­well was an in­tel­lec­tu­al gi­ant. But he was mod­est and un­as­sum­ing on a per­son­al level. He al­ways dressed prop­erly in an aged jack­et/suit, and he drove an old car that of­ten in­vited jokes from stu­dents. While gradu­ate stu­dents all over the world were learn­ing about the Rao–Black­well the­or­em, I nev­er saw him teach­ing a gradu­ate course. He en­joyed teach­ing un­der­gradu­ate courses, and he placed great em­phas­is upon spend­ing time on pre­par­a­tion to im­prove classroom teach­ing.

A great mind and a great spir­it has de­par­ted. The world is a rich­er place be­cause of his writ­ings, but those of us who had the priv­ilege of meet­ing him per­son­ally have be­nefited even more.