Celebratio Mathematica

David H. Blackwell

A Tribute to David Blackwell

by Persi Diaconis

Dav­id Black­well was a bril­liant, gentle man. I don’t think I ever met any­one with so many IQ points fused in­to such an agree­able ex­ter­i­or. We had many areas of con­tact: Bayesian stat­ist­ics, de­script­ive set the­ory, and his damned tri­angle prob­lem. Let me briefly com­ment on each.

Dav­id had been con­ver­ted (his phrase) to the Bayesian view dur­ing a walk with Jimmy Sav­age. Bayes made sense to Dav­id, and he made sense of it to oth­ers. Since not every­one is a Bayesian, some of us learn to speak both clas­sic­al and Bayesian lan­guages. Once, Dav­id heard me give a col­loqui­um us­ing both lan­guages. Af­ter­ward, he gave me a really hard time! “Per­si, it soun­ded as if you were apo­lo­giz­ing for be­ing a Bayesian. You don’t have to apo­lo­gize, it’s the only sens­ible stat­ist­ic­al the­ory.”

Berke­ley was a hot­bed of de­script­ive set the­ory — ana­lyt­ic, coana­lyt­ic, and uni­ver­sally meas­ur­able sets were friends of Black­well, Du­bins, and Freed­man. I learned the sub­ject from Freed­man and wrote a small pa­per with Black­well: the Amer­ic­an Math­em­at­ic­al Monthly had pub­lished a longish pa­per con­struct­ing a non­meas­ur­able tail set in coin-toss­ing space. We no­ticed that the stand­ard con­struc­tion of a non­meas­ur­able set due to Vi­tali also had this prop­erty. We sent a one-page note to the Monthly and re­ceived a scath­ing ref­er­ee’s re­port in re­turn: “Why would you send a pa­per about such junk to the Monthly?” Clearly, not every­one likes meas­ure the­ory. We pub­lished our pa­per else­where. I re­cently gave a talk on it at Hal­loween (when the mon­sters come out).

I had din­ner with Dav­id, Erich Lehmann, Ju­lie Shaf­fer, and Susan Holmes about a year be­fore he died. You don’t ask someone in his high eighties if he’s still think­ing about math. However, Dav­id had taken up com­put­ing late in life, and I asked if he was still at it. He answered with a loud pound on the table and “Yes, and dam­nit I’m stuck.” He ex­plained: “Take any tri­angle in the plane. Con­nect the three ver­tices to the mid­points of the op­pos­ite sides; the three lines meet in the middle to give the bary­centric sub­di­vi­sion in­to six tri­angles. If you do it again with each of the six little tri­angles, you get thirty-six tri­angles, and so on.” He no­ticed that most of the tri­angles pro­duced get flat (the pro­por­tion with largest angle great­er than 180 de­grees minus ep­si­lon tends to one). He was try­ing to prove this and had got­ten stuck at one point. His proof idea was bril­liant: he turned this geo­metry prob­lem in­to a prob­ab­il­ity prob­lem. Con­struct a Markov chain on the space of tri­angles by pick­ing one of the six in­side, at ran­dom. He defined a func­tion on tri­angles that was zero at per­fectly flat tri­angles and that he be­lieved was su­per­har­mon­ic. Non­stand­ard the­ory shows the it­er­a­tion tends to zero; hence the av­er­age tends to zero, and so “most tri­angles are flat”. The proof of su­per­har­mon­icity was a tri­go­no­met­ric night­mare, and he had not been able to push it through. I couldn’t be­lieve that such a simple fact about tri­angles was hard (it was). I tricked sev­er­al col­leagues in­to work­ing it through by vari­ous com­plex ar­gu­ments. After a lot of work, we found the res­ult in a lovely pa­per by Barany, Bear­don, and Carne (1996). Black­well’s ap­proach re­mains a tan­tal­iz­ing pos­sib­il­ity.