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.
