Excerpt from Statistical Monitoring of Clinical Trials
Einstein developed the notion that a statistical treatment of complex systems could provide useful insight into the behavior of complex systems (like the trillions of molecular bombardments that a microscopic pollen granule sustains in suspension). However it was difficult to do anything of practical importance with this result, primarily because the mathematics of probability had not yet been sufficiently developed. However, this last giant step that moved this field toward the application of clinical science was undertaken by a mathematical prodigy, Norbert Wiener.
The story of Norbert Wiener is not just the demonstration of the power and idiosyncrasies of the mathematical personality. It is a story of remarkable, parental persistence and optimism.
Norbert Wiener’s father, Leo Wiener was a physician and an engineer in Europe. When he emigrated to the United States, like many immigrants, he was not permitted to practice his trade. He became a farmer, a factory worker, and a teacher. However, he was always interested in mathematics [[i]].
His son, Norbert above all wanted to read, becoming a voracious reader at an early age. However, he was not good in arithmetic and had tremendous difficulty in school. In his own words later in life, Norbert Wiener said “My chief deficiency was mathematics”[[ii]]. His father, responded by removing him from school, taking on the major role of tutoring his son. However, Norbert was a difficult and trying student who had no affection for arithmetic. Compounding this difficulty was the boy’s physical awkwardness and clumsiness. Norbert’s poor coordination, in concert with profound and poorly corrected nearsightedness, made it impossible for him to carry out even the simplest physical demonstration for himself. His father persisted, working with his son over the next few months. The months turned into years.
The result was miraculous. Six years after he professed a crippling and limiting weakness in mathematics, Norbert Weiner was admitted to Tufts College at the age of eleven. He completed his undergraduate studies when he was fourteen, upon which time he was admitted to Harvard graduate school. Pursuing training first in zoology, and then mathematics he completed his Ph.D. at 18. Shortly thereafter, he became a mathematics instructor at MIT, where he remained for his entire career. He was 24 years old.
The first problem to which Norbert Weiner turned his attention was the mathematics of Brownian motion. Much had happened in probability since Einstein’s 1905 publication fifteen years earlier. Wiener was intrigued by the possibility that Einstein’s work opened. Was is possible to compute the probability that a particle would follow a pre-specified trajectory? Weiner recognized that it would be impossible to predict the precise path that a microscopic particle being bombarded by trillions of molecules every moment along the way would follow. The number of equations that would need to be solved would be far beyond the calculating ability available. However, he reasoned, it might be possible to determine the likelihood of certain trajectories or paths. For example, what is the probability that a particle, after a time period of one second, has moved at least 1 unit up from its starting point? There were many trajectories the particle could follow. A fraction of those would meet this criteria. How likely was this fraction?
Many eminent mathematicians attempted to solve this problem, attempting to link Einstein’s theoretical work to this practical application.[1] Wiener actually did it, by creating 1) a new concept, the stochastic process, and 2) an innovative measure that still bears his name – Wiener measure. Wiener measure permits the accurate computation of these excursion probabilities that are critical to modern statistical monitoring procedures in clinical research. Essentially, clinical trial methodologists model the movement of a test statistic much like Einstein and now Wiener modeled the movement of particles, using the normal distribution.
Wiener was somewhat of an eccentric. His speech, like his writings were difficult to understand. All too often, Wiener was unable to resist the temptation to say everything that came into his mind. He could not separate mathematics from is implications, nor its implications from his personal experiences. It was as though in Wiener’s mind, the person he was addressing instantaneously changed from a layman, to a mathematician, and then to Wiener himself [[iii]].
It was the first formal demonstration that probability theory could be applied to events that occur randomly in time, and created the new study of stochastic processes, This field was essentially a creation of Norbert Wiener. The mathematical tools were now available to compute probabilities of trajectories.
[1] Including the likes of Borel, Lebesque, Lévy, Banach, Fréchet, and even the preeminent mathematician, A. N. Kolmogorov.
[i]. School of Mathematics and Statistics. University of St. Andrews, Scotland. http://www-gap.dcs.st-nd.ac.uk/~history/Mathematicians/Wiener_Norbert.html
[ii] Norbert Wiener, (1966). Bull. Amer. Math. Soc. 72.
[iii]. Freudenthal, H. Biography of Norbert Wiener. Dictionary of Scientific Biography (New York 1970-1990).