by Peter Bickel
I first met David Blackwell when I took his course on information theory during my first year as a doctoral student. David had chosen as a text Jack Wolfowitz’s Information Theory for Mathematicians, which, as the title suggests, was somewhat dry. David made the subject come to life. His style was well established. Strip the problem of all excess baggage and present a solution in full elegance. The papers that I read of his, such as those on the Blackwell renewal theorem and on Bayesian sequential analysis/dynamic programming, all have that character. I didn’t go on in information theory, but I didn’t foreclose it. My next memorable encounter with David, or rather the strength of his drinks, was at a party he and Ann gave for the department. When I declined his favorite martini he offered Brandy Alexanders. I took two and have trouble remembering what happened next!
And then I had the great pleasure and good fortune of collaborating with David. I was teaching a decision theory course in 1966, relying heavily on David and Abe Girshick’s book, Theory of Games and Statistical Decisions. I came across a simple, beautiful result of theirs that, in statistical language, can be expressed as: If a Bayes estimator is also unbiased, then it equals the parameter that it is estimating with probability one. In probabilistic language this says that if a pair of random variables form both a forward and a backward martingale, then they are a.s. equal.
Unbiasedness and Bayes were here specified in terms of squared error loss. I asked the question “What happens for \( L_p \) loss for which a suitable notion of unbiasedness had been introduced by Lehmann?” I made a preliminary calculation for \( p \) between 1 and 2 that suggested that the analogue of the Blackwell–Girshick result held. I naturally then turned to David for confirmation. We had essentially an hour’s conversation in which he elucidated the whole story by giving an argument for what happened when \( p \) equals 1, and, in fact, the result failed. He then sent me off to write it up. The paper appeared in 1967 in the Annals of Mathematical Statistics.
It is still a paper I enjoy reading. It led to an interesting follow-up. In a 1988 American Statistician paper, Colin Mallows and I studied exhaustively what happens when the underlying prior is improper, which led to some surprises. David was a Bayesian belonging, I think, to the minority who believed that axioms of rational behavior inevitably lead to a (subjective) prior. He was essentially alone in that point of view in the department but never let his philosophical views interfere with his most cordial personal relations.
Sadly, our collaboration was the last of my major scientific contacts with David. We were always on very friendly terms, but he would leave the office at 10 AM, which was my usual time of arrival.
After we both retired, we would meet irregularly for lunch at an Indian restaurant, and I got a clearer idea of the difficulties as well as triumphs of his life. Despite having grown up in the segregated South, David always viewed the world with optimism. As long as he could do mathematics, “understand things”, rather than “doing research”, as he said in repeated interviews, he was happy.
It was my fortune to have known him as a mathematician and as a person. He shone on both fronts.
