Celebratio Mathematica

David H. Blackwell

A Tribute to David Blackwell

Francisco J. Samaniego

Be­cause of Dav­id Black­well’s widely re­cog­nized geni­us, as evid­enced in his path-break­ing re­search, the many cre­at­ive ideas he gen­er­ously shared with stu­dents and col­leagues, his elec­tion to the Na­tion­al Academy of Sci­ences, and his re­ceipt of the coveted Berke­ley Cita­tion upon his form­al re­tire­ment from the fac­ulty, it is per­haps un­der­stand­able that an­oth­er fa­cet of his re­mark­able ca­reer would be less known and less uni­ver­sally cel­eb­rated. This fa­cet was his ex­traordin­ary abil­ity to teach math­em­at­ics and stat­ist­ics in new, clear, and com­pel­ling ways. There is a good deal of evid­ence that may be ad­vanced in sup­port of the pro­pos­i­tion that Black­well was an ex­cep­tion­al teach­er. We would be re­miss if this as­pect of his won­der­fully suc­cess­ful ca­reer was over­looked in the present over­view of his work. While our dis­cus­sion of Black­well’s teach­ing is ne­ces­sar­ily brief, we hope that we will leave no doubt among read­ers of this piece that Black­well was a pree­m­in­ent teach­er and ment­or.

The most telling evid­ence of Black­well’s teach­ing prowess is simply the testi­mo­ni­als from stu­dents and col­leagues that ex­ist in a num­ber far too great to at­tempt a com­pre­hens­ive sum­mary. Suf­fice it to say that many of his stu­dents con­sidered him to be the finest in­struct­or that they ever had the priv­ilege to study with. His style was un­fail­ingly en­ga­ging, as it was his cus­tom to share his nat­ur­al curi­os­ity with his stu­dents, ex­plain­ing not just the “how” as­so­ci­ated with a stat­ist­ic­al pro­ced­ure but the “why” as well, along with the mo­tiv­a­tion for the ideas in­volved and the (of­ten sur­pris­ing) con­nec­tions with oth­er ideas usu­ally of in­terest in their own right. It was a pleas­ure to hear him speak. One gen­er­ally came away from a lec­ture by Dav­id Black­well both im­pressed with his mas­tery of the sub­ject and in­trigued by ques­tions he had left his audi­ence to think about. His col­leagues at Berke­ley looked to him as a mod­el and of­ten sought his ad­vice on the best way to present a giv­en top­ic (as well as on a host of oth­er mat­ters, per­son­al and pro­fes­sion­al). In spite of his won­der­ful gifts as a teach­er, Black­well was very mod­est about his skills and would give his ad­vice as if it was a tent­at­ive, off-the-cuff sug­ges­tion. Once, in a re­cep­tion pri­or to a sem­in­ar he presen­ted at UC Dav­is, a former stu­dent of his asked him, “Dav­id, what do you do when you’ve presen­ted an idea in a way that you con­sider to be ‘just right’, and a stu­dent raises his hand and says ‘I didn’t get that’?” Without miss­ing a beat, Dav­id answered, “Well, I just re­peat ex­actly what I just said, only louder.”

Dav­id Black­well’s well-honed teach­ing in­stincts were as evid­ent in his writ­ings as they were in the classroom. Two of his many pub­lished “notes” come to mind in this re­gard. These notes ap­peared in the An­nals of Stat­ist­ics volume in which Thomas Fer­guson presen­ted his now cel­eb­rated pa­per on Bayesian non­para­met­rics (an idea, by the way, that he ac­know­ledged as groun­ded in his dis­cus­sions with Black­well). In a note en­titled “Dis­crete­ness of Fer­guson Se­lec­tions”, Black­well gave an ele­ment­ary proof of the dis­crete­ness of draws from a Di­rich­let pro­cess, shed­ding much light on this par­tic­u­lar char­ac­ter­ist­ic of Di­rich­let pro­cesses (which had been proven by Fer­guson in an An­nals of Prob­ab­il­ity pa­per us­ing much more com­plex ar­gu­ments). In the same AoP is­sue, Black­well and Mac­Queen presen­ted an al­tern­at­ive de­riv­a­tion of the Di­rich­let pro­cess us­ing a lovely and quite in­tu­it­ive con­struc­tion in­volving Pólya urn schemes. The lat­ter pa­per has led to much fruit­ful re­search in Bayesian non­para­met­rics. Both pa­pers con­tained use­ful tech­niques, but their greatest con­tri­bu­tion was, without doubt, the cla­ri­fic­a­tion of the prop­er­ties and po­ten­tial of Fer­guson’s Di­rich­let pro­cess.

Black­well pub­lished the ele­ment­ary text­book Ba­sic Stat­ist­ics in 1969. The book is unique in the field and is re­com­men­ded read­ing both for stu­dents just be­ing ex­posed to the sub­ject and, we dare say, for the stat­ist­ics com­munity as a whole. It is no ex­ag­ger­a­tion to refer to the book as a “gem”. In the book, Black­well covered the “stand­ard top­ics” found in an in­tro­duct­ory course — ele­ment­ary prob­ab­il­ity, the bi­no­mi­al and nor­mal mod­els, cor­rel­a­tion, es­tim­a­tion, pre­dic­tion, and the chi-square test for as­so­ci­ation. The treat­ment of these top­ics was, however, fresh and crisp, with most of the ideas mo­tiv­ated by think­ing about draw­ing balls from urns. For ex­ample, he chose to in­tro­duce the idea of Bayesian point es­tim­a­tion through the prob­lem of es­tim­at­ing the num­ber of fish in a pond via a mark-re­cap­ture ex­per­i­ment. Al­though the math­em­at­ic­al level of the book was in­ten­tion­ally low, the con­cep­tu­al reach was much broad­er than what one usu­ally finds at the in­tro­duct­ory level. In his pre­face, Black­well de­scribes his ap­proach as “in­tu­it­ive, in­form­al, con­crete, de­cision-the­or­et­ic and Bayesian”. He took on the no­tions of prob­ab­il­ity dens­it­ies, mean squared er­ror, mul­tiple cor­rel­a­tion, pri­or dis­tri­bu­tions, point es­tim­a­tion, and the nor­mal and chi-square ap­prox­im­a­tions, all with the very mod­est ex­pect­a­tion that the stu­dents read­ing the book “could do arith­met­ic, sub­sti­tute in simple for­mu­las, plot points and draw a smooth curve through plot­ted points”. He was true to his prom­ise of mak­ing stat­ist­ics ac­cess­ible to any­one who had only these skills. Per­haps the most re­mark­able thing about this book is that Black­well man­aged to pack a treas­ure trove of ideas in­to 138 pages, di­vided in­to six­teen chapters and con­tain­ing 118 prob­lems and their solu­tions. He had a gift for get­ting to the core of the top­ics he wrote or taught about. This book is a lovely ex­ample of that gift in ac­tion.