He studied mathematics at the University of California, Berkeley, doing a premedical programme at the same time.
Birnbaum's thesis and his early work was very much in the spirit of Lehmann's classic text Testing Statistical Hypotheses.
Birnbaum stayed at Columbia until 1959 when he moved to the Courant Institute of Mathematical Sciences, becoming a full Professor of Statistics in 1963.
The article in the Leading Personalities volume opens with the declaration, "Allan Birnbaum was one of the most profound thinkers in the field of foundations of statistics."
Leonard Jimmie Savage opened the discussion by saying Without any intent to speak with exaggeration or rhetorically, it seems to me that this is really a historic occasion.