Ulf Grenander was born in 1923 in Vastervik, Sweden, a small coastal town on the Baltic Sea. His degrees were from Uppsala University (B.A., 1946; Licentiate of Philosophy, 1948) and Stockholm University, where he studied under the great statistician Harald Cramér and received his PhD in 1950.
Grenander traced his interests in probability and statistics to a childhood fascination with quantum mechanics. As an undergraduate, he was influenced by the former code breaker, and well-known analyst, Arne Beurling, and later by Harald Bohr, a mathematician and brother of Niels Bohr. Following his graduation, Grenander served a year in the military, about which he said, “Military service was not so bad! I discovered the pleasures of outdoor life instead of being a nerd.”
His graduate years culminated with his landmark dissertation on the mathematical foundations of statistical inference for continuous parameter stochastic processes. This drew the attention of Kolmogorov, among others, and many invitations followed. Grenander spent the 1951–52 academic year at the University of Chicago. He shared an office with Charles Stein and rented Jimmy Savage’s apartment. He met many prominent statisticians during his year in Chicago, including Bill Kruskal, Murray Rosenblatt, Leo Goodman and Joe Hodges.
Grenander and Rosenblatt struck up a collaboration that led to their influential Statistical Analysis of Stationary Time Series. While at Chicago, Jerzy Neyman invited him to spend the 1952–1953 year at Berkeley, where he met Gábor Szegő at a joint seminar with Stanford University. The two shared an interest in Toeplitz forms, and over the next few years they wrote their well-known book Toeplitz Forms and Their Applications.
Grenander spent most of the years 1953–66 at the University of Stockholm, eventually becoming the Director of the Institute for Insurance Mathematics and Mathematical Statistics. During these years he made many more seminal contributions to inference and indulged an abiding interest in real-world problems by consulting for the insurance industry. His applied work had a lasting effect on the actuarial sciences, where he offered analytic solutions to replace empirical tables and used nonparametric maximum likelihood methods to more accurately predict payouts. He spent the 1957–58 academic year visiting the Division of Applied Mathematics at Brown University, at the invitation of William Prager, who had been impressed with Grenander’s work on Toeplitz forms. There, he taught probability and statistics, and rekindled a long-standing fascination with computing, thanks largely to the excellent computing resources donated by IBM. Although the Division was almost entirely focused on mechanics and dynamical systems, Grenander returned in 1966 and remained there, as the L. Herbert Ballou University Professor, until his retirement.
In the early 1960s, while still in Sweden, Grenander began a new research program, seeking more general and abstract formulations of statistical models. His 1963 monograph on Probabilities on Algebraic Structures explored the mathematical foundations for probability distributions on “regular structures,” and was the beginning of his effort to produce a general theory of patterns. In his 1981 monograph on Abstract Inference, he developed new nonparametric models and methods, including the “Method of Sieves.” These new directions became the foundations of a remarkable marriage of combinatorial and stochastic structure that he called “pattern theory.” Pattern theory blossomed at Brown, aided by Walter Freiberger and the eventual hiring of new faculty in probability and statistics.
By the 1980s, Grenander had begun to explore applications of the theory, including image restoration, image synthesis and analysis, language processing, and even musical composition. In an especially fruitful application, Grenander and collaborators undertook the construction of a library of models for the structures of the human body—a “digital anatomy” equipped with measures of normal shape and normal variability.
Many of today’s leaders in the quantification of biological shapes and their distributions were drawn to the field by Grenander’s elegant and rigorous formulations. In an interview he remarked, “I refer to pattern theory as the intellectual adventure of my life.”
Grenander published over 90 research papers and 15 authored books, and earned many honors and awards, including Fellow of the Institute of Mathematical Statistics (1953), Member of the Royal Swedish Academy of Science (1965) and Honorary Fellow of the Royal Statistical Society, London (1989). He delivered numerous prestigious lectures, received an honorary Doctorate of Science degree from the University of Chicago (1993), and was a Fellow of the American Academy of Arts and Sciences (1995) and a member of the National Academy of Sciences, USA (1998). In recognition of his path-breaking contributions, the American Mathematical Society has established the endowed Ulf Grenander Prize in Stochastic Theory and Modeling.
Grenander’s diverse scientific interests are reflected in the breadth of his influence. His seminal thesis under Cramér, together with his later collaboration with Rosenblatt, delivered a rigorous mathematical and statistical framework for signal processing. New methods for estimation and hypothesis testing for stationary stochastic processes emerged and were widely adopted by the engineering community. His work with Szegő on Toeplitz forms found numerous applications, including, for example, in the study of random fields, speech recognition, state-space modelling, model selection and information theory. His actuarial work remains fundamental to the insurance industry. And his work in abstract inference demonstrated, constructively, that virtually any decision function, density or regression surface, parametric or not, could be systematically estimated given enough data, anticipating the utility of big data and its applications to machine learning. In the context of Bayesian inference, he argued for and demonstrated the potential of Monte Carlo methods, seeding a revolution in statistics and artificial intelligence. And his pattern theory led many of us to the conviction that structured probabilistic models would be fundamental to the most advanced technologies of the future, inspiring David Mumford’s prediction that the 21st century would be the “Age of stochasticity.”
Grenander was a voracious reader, broadly knowledgeable in history and science, and fluent in many languages. He was a passionate sailor and a skilled “do-it-yourself” electrician, plumber and carpenter. Almost single handedly he built entire wings of his summer home in Vastervik, which now accommodate frequent stays by his three children and six grandchildren, all of whom, in addition to his wife Emma-Stina, survive him.
Written by Stuart Geman, Brown University