Statistics - David Blackwell

DAVID BLACKWELL - Scholar of Probability

“I'm interested in understanding…and often to understand something you have to work it out yourself because no one else has done it.”

— David Blackwell

The first African American inducted into the National Academy of Sciences, and the first black tenured faculty member at UC Berkeley. At the age of 22, he was awarded a PhD in mathematics by the University of Illinois.

Blackwell was a pioneer in textbook writing and game theory. Blackwell wrote one of the first Bayesian textbooks (Basic St David Blackwell, a statistician and mathematician who wrote groundbreaking papers on probability and game theory and was the first black scholar to be admitted to the National Academy of Sciences.

As a consultant to the RAND Corporation from 1948 to 1950, he applied game theory to military situations. It was there that he turned his attention to what might be called the duelist’s dilemma, a problem with application to the battlefield, where the question of when to open fire looms large.

He was hired by UC Berkeley as a full professor in the newly created Statistics Department in 1955, becoming the Statistics department chair in 1956.

His “Basic Statistics” (1969) was one of the first textbooks on Bayesian statistics, which assess the uncertainty of future outcomes by incorporating new evidence as it arises, rather than relying on historical data. He wrote numerous papers on multistage decision-making.

The work of David Blackwell known for;

Rao–Blackwell theorem Blackwell's approachability theory

Blackwell channel Sequential analysis

Bayes’ Theorem

David Blackwell’s work was in the area of inferential statistics. If you have no idea about something and you’re trying to reach some conclusion about it, if you receive some additional accurate information about it, your second conclusion will never be more inaccurate than your original conclusion.

The usefulness of Bayesian thinking is when the additional piece of information is leading you in proper direction but has the possibility of being misleading.

Stated mathematically the probability of A given B (that second piece of information)

P(A|B) = (P(B|A)) x P(A)/P(B)

Where A and B are events P (B) is not equal to 0

P (A) and P (B) are the probabilities of observing A and B without regard to each other.

P (A|B), a conditional probability, is the probability of observing event A given that B is true.

P (B|A) is the probability of observing event B given that A is true.

MATH PROBLEM: Prisoners Dilemma

The approach David Blackwell’s work will be to set-up a Prisoners’ Dilemma scenario such that one person will play against three different other competitors, for 10 simultaneous iterations of three players each.

Corresponding Prisoner A - will follow the Tit-for-Tat strategy with a cooperating opening position. Whatever the test player does, the on the previous move, prisoner A will do on the following turn.

Corresponding Prisoner B - will follow a Tit-for-Tat strategy with a cooperating opening position, with the following exceptions:

For the first 10 iterations, random deviation and learning will be, represented by a throw of a die. If the die is 1-4 the person follows Tit-for-Tat. If a 5 or a 6 is thrown then the person throws the die again and follows tit-for-tat if the second throw is even and takes the opposite position if the second throw is odd.

For the second 10 iterations, the above process will repeat except only triggered by a six on the initial throw.

For the third 10 iterations, the player will follow a tit-for-tat strategy.

Corresponding Prisoner C - the remainder of the students taking individual turns taking each position as that person chooses.

After thirty iterations of the game, the homework task will be to see how lone if at all, the test subject took to figure out how to maximize their benefits.