Dec 21, 2009

David Blackwell

Given the trip to NIPS and the holiday period, posting has been quite slow. So, now that things are starting to pick up, I decided to sit down and write about Blackwell's Approachability Theorem, which pops up in a number of game theory and prediction problems. But after doing a bit of internet research on the theorem, I decided that the result's author, David Blackwell, is so interesting that he deserves his own post.

Blackwell is an impressive but relatively unknown figure of mid-20th century mathematics, and he's currently an emeritus professor in the Statistics department at my own UC Berkeley. (Chritos Papadimitriou claims to have run into him at one point) Blackwell has a rather limited
personal web page, a Wikipedia page that is only slightly more informative, and a handful of articles/profiles written on him like this one - a surprisingly-sparse web presence given his accomplishments.

Blackwell is perhaps most well-known for the
Rao-Blackwell theorem, a staple of any standard theoretical statistics class. Rao and Blackwell showed that, given some parameterized family of distributions, any unbiased estimator of one such parameter can always be "improved" by taking its conditional expectation with respect to any "sufficient statistic" of the parameter. This is indeed a surprising result as it provides a simple procedure for transforming any unbiased estimator into a "better" one, i.e. one having lower variance.

But Blackwell's work was not restricted to Statistics. The Approachability Theorem mentioned above is purely a Game Theory result, and essentially extends Von Neumann's Minimax Theorem. And it goes further - his publication record spans "Baysian statistics, probability, game theory, set theory, dynamic programming, and information theory."

Finally, while certainly irrelevant with respect to his capabilities, it requires mentioning that David Blackwell is an African-American. Recall: his career developed at a time when Jim Crow Laws were still in existence. His race would have been no minor obstacle. It
appears, however, that he took it all in stride:

While Blackwell was growing up, "Southern Illinois was probably fairly racist. But I was not even aware of these problems -- I had no sense of being discriminated against." At 16 he enrolled at the University of Illinois and expected to be an elementary school teacher. There were no black faculty members on that campus. In 1943, after two years as a prestigious postdoctoral fellow of the Institute for Advanced Study, Princeton, he wrote 104 letters of application for university teaching jobs -- all to black colleges. "I just assumed that [the door] was closed." He spent the next ten years at Howard University, building a national reputation as a creative researcher and a talented teacher.

By the 1950's the employment situation for blacks had begun to change. "I feel that there must have been a big change just resulting from World War II." In 1954 Blackwell accepted an invitation to join the faculty of the newly formed Department of Statistics (separate from the Department of Mathematics) at the University of California, Berkeley. He enjoys "being part of a smaller group.....where everybody talks to everybody else." Although he has taken his turn chairing the department, he does not miss the administrative role. "When I gave up being chairman, for about a year my first waking thought was, 'I'm no longer chairman,' and it made my day."