A Sabbath-keeping boy in Kolozsvár
Abraham Wald was born on October 31, 1902, in Kolozsvár, a university city in Transylvania that was then part of the Kingdom of Hungary and is today Cluj-Napoca in Romania. He came from a religious Jewish family of intellectuals who, like many Jews in the region, had been pushed into trades well below their abilities; his grandfather was Rabbi Moshe Shmuel Glasner, a prominent Hungarian rabbi. Hungarian schools then required attendance on Saturdays, which the family would not accept, so Wald did not go to primary or secondary school at all. His parents, both capable teachers, educated him at home.
The arrangement evidently suited him. When he did enter formal education, at the university in Cluj, by then part of Romania after the First World War, it was as a gifted mathematician, though contemporaries recorded that admission was not made easy for a Jewish student. He graduated in mathematics in 1928.
Vienna and the Menger circle
In 1927 Wald went to the University of Vienna to study under the mathematician Karl Menger, son of the economist Carl Menger, and earned his doctorate in 1931 with work in geometry. Menger ran a celebrated mathematical colloquium that drew some of the sharpest minds in Europe, and Wald quickly became one of its stars, publishing twenty-one papers on geometry between 1931 and 1937.
A university post, however, was out of the question. Austrian academic life was closed to young Jewish scholars, and Menger’s own seminar was eventually forced to stop, in part because of criticism of its Jewish contributors. Wald made his living as mathematics tutor to the banker and economist Karl Schlesinger, and the economist Oskar Morgenstern, who directed the Austrian Institute for Business Cycle Research, found him a position there. The detour into economics proved fruitful: Wald produced some of the first mathematically rigorous proofs that a competitive economic equilibrium could exist, and a 1936 monograph on seasonal patterns in time series that Morgenstern judged superior to all other techniques of the day.
Wald kept working in economics after he reached America. In 1939 he published in Econometrica a new formula for the cost-of-living index, and in 1940 a paper on fitting a straight line when both variables are measured with error, a practical problem in every experimental science. These papers show a mathematician who, despite his abstract gifts, was repeatedly drawn to questions that people outside mathematics actually needed answered.
Escape, and a family destroyed
When Nazi Germany annexed Austria in March 1938, the discrimination Wald had long endured turned into open persecution. He left Vienna that year on an invitation from the Cowles Commission for Research in Economics in the United States, and by September 1938 he was at Columbia University in New York as a Carnegie Corporation fellow, learning modern statistics from Harold Hotelling. He began lecturing at Columbia in 1939, joined its faculty in 1941 and stayed there for the rest of his life.
Most of his family could not follow. According to the University of St Andrews’ MacTutor biography, of the nine members of his family he had left behind, all but one were murdered in the gas chambers of Auschwitz. In America he met and married Lucille Lang; they had two children, one of whom, Robert Wald, became a distinguished theoretical physicist.
The planes that didn’t come back
During the Second World War, Wald joined the Statistical Research Group at Columbia, a small team of mathematicians working on military problems for the US government. There was an absurd obstacle at first: as a recent arrival he was formally an enemy alien, barred from seeing the classified material he was meant to analyse. Hotelling later recalled the joke that each page Wald wrote should be snatched away and never shown to him again; the problem was solved when a federal court heard his case early and promptly naturalised him.
Among the group’s problems was how to protect bombers from enemy fire. In a series of eight memoranda written in 1943, Wald developed a method for estimating how vulnerable each part of an aircraft was, using only the damage seen on planes that survived. His key insight was that the sample was biased: if hits were spread roughly evenly over an aircraft, then parts that rarely showed damage on returning planes were the parts where a hit was most likely to bring a plane down. The bullet holes marked where a bomber could be hit and still come home. This is the origin of the most famous illustration of what is now called survivorship bias.
Historians add a caution. The popular version, officers proposing to armour the holes, Wald overruling them with a single dramatic remark, is a later retelling; the surviving record is Wald’s technical memoranda, which estimate vulnerability rather than issue instructions. But those memoranda were real and influential. The Center for Naval Analyses reprinted them in 1980, and the method was used to analyse aircraft damage in the Korean and Vietnam wars.
Deciding when you have seen enough
Wald’s most important wartime contribution was arguably less glamorous. Military factories needed to test munitions and equipment without wasting time or materiel, and Wald asked a simple question: instead of fixing the number of tests in advance, why not check the evidence after each one and stop as soon as it is conclusive? The answer was the sequential probability ratio test, the foundation of a new field he called sequential analysis. It often reached reliable decisions with far fewer observations than classical methods. Wald conjectured its optimal properties in 1943, proved them with his colleague Jacob Wolfowitz in 1948, and set out the theory in his book Sequential Analysis (1947).
Behind both projects lay a grander idea. In a 1939 paper and then in his book Statistical Decision Functions (1950), Wald recast the whole of statistics, testing hypotheses, estimating quantities, as special cases of one problem: how to make a decision under uncertainty when mistakes have costs. Wolfowitz called the 1939 paper probably Wald’s most important single work. Statistical decision theory remains one of the pillars of the modern discipline.
What his ideas do today
Wald’s name is attached to an unusual number of everyday tools: the Wald test, used in countless regression analyses; the Wald–Wolfowitz runs test; Wald’s equation in probability; the Wald distribution. Sequential analysis made industrial quality control cheaper and faster, and its descendants are now used to monitor clinical trials, allowing researchers to stop a trial early when a treatment is clearly working or clearly harmful, saving time, money and sometimes lives. The same logic sits behind the continuous A/B testing that technology companies run every day.
His bomber analysis, meanwhile, has become a parable taught far beyond statistics: in business schools, medicine, engineering and journalism, as a warning that the data in front of you may be shaped by what is missing. Few mathematicians have had a single idea travel so widely, or so usefully, into ordinary reasoning.
His decision theory also introduced the “maximin” approach to choosing under deep uncertainty: when probabilities cannot be trusted, pick the option whose worst possible outcome is least bad. Wald’s maximin model remains a standard reference point in operations research, robust optimisation and the economics of risk, wherever planners must guard against the worst case rather than bet on the average.
A quiet and gentle man
Those who knew Wald remembered him as modest and absorbed. A student who attended his lectures described him as “a quiet and gentle man, deeply immersed in his work.” He supervised doctoral students who themselves became leading statisticians, including Herman Chernoff and Charles Stein, and helped make Columbia one of the world’s centres of mathematical statistics.
He was also a mathematician to his core, which sometimes drew criticism. After his death the British statistician Ronald Fisher attacked him as a pure mathematician without experimental experience. Jerzy Neyman published a defence of Wald’s work the following year, and later theorists such as Lucien Le Cam credited Wald’s writings as the leading influence on their own.
Death in the Nilgiri Mountains
In 1950 the Government of India invited Wald on an extended lecture tour. He visited the Indian Statistical Institute in Calcutta and was due to attend the Indian Science Congress in Bangalore in January. On December 13, 1950, the Air India DC-3 carrying him and his wife crashed in the Nilgiri Mountains in southern India, killing them both. Wald was forty-eight. Their two children were at home in the United States.
He had lived barely twelve years in America, and his most productive period was cut short just as the theory he had built was being taken up across the sciences.
Legacy
Abraham Wald’s life traced a path familiar from twentieth-century Jewish history: a religious childhood in Central Europe, brilliance blocked by discrimination, flight from Nazism, the murder of most of his family, and a second life in America that he spent, in part, helping the country that had taken him in to win a war. His intellectual legacy is unusually broad, in pure mathematics, economics, statistics and operations research, and unusually practical.
Every clinical trial that stops early because the evidence is already clear, every analyst who asks what is missing from a dataset, every engineer who thinks about the planes that did not return, is drawing on his work. The legend of the bombers may have grown in the retelling, but the habit of mind it teaches is exactly his.
Sources & further reading
Abraham Wald, Wikipedia · Abraham Wald, MacTutor History of Mathematics, University of St Andrews · The Legend of Abraham Wald, American Mathematical Society Feature Column · Lessons from World War II Statisticians: Survivorship Bias and Sequential Analysis, Conversable Economist · Researched from the sources above and checked against them. Spotted an error? Tell us.
A life of tikkun olam. This is part of tikkun olam, the Jewish idea of repairing the world, and every one of us can add to it.