Bernard Koopman
Encyclopedia
Bernard Osgood Koopman was a French-born American mathematician, known for his work in ergodic theory
, the foundations of probability, statistical theory
and operations research
.
After living in France and Italy, the Koopmans immigrated to the United States in 1915. Koopman was a student of George David Birkhoff
and his initial work concentrated on dynamical system
s and mathematical physics
.
The Pitman–Koopman–Darmois theorem states that the only families of probability distributions that admit a sufficient statistic whose dimension remains bounded as the sample size increases are exponential families
.
Koopman's mother was a first cousin of William Fogg Osgood
.
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, the foundations of probability, statistical theory
Statistical theory
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that...
and operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...
.
After living in France and Italy, the Koopmans immigrated to the United States in 1915. Koopman was a student of George David Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
and his initial work concentrated on dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
.
The Pitman–Koopman–Darmois theorem states that the only families of probability distributions that admit a sufficient statistic whose dimension remains bounded as the sample size increases are exponential families
Exponential family
In probability and statistics, an exponential family is an important class of probability distributions sharing a certain form, specified below. This special form is chosen for mathematical convenience, on account of some useful algebraic properties, as well as for generality, as exponential...
.
Koopman's mother was a first cousin of William Fogg Osgood
William Fogg Osgood
William Fogg Osgood was an American mathematician, born in Boston.In 1886, he graduated from Harvard, where, after studying at the universities of Göttingen and Erlangen , he was instructor , assistant professor , and thenceforth professor of mathematics...
.
Publications
- "Hamiltonian SystemHamiltonian systemIn physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
s and Transformations in Hilbert SpaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
" Proceedings of the National Academy of SciencesProceedings of the National Academy of SciencesThe Proceedings of the National Academy of Sciences of the United States of America, usually referred to as PNAS, is the official journal of the United States National Academy of Sciences...
17, 315–318, 1931. - "On distributions admitting a sufficient statistic. Transactions of the American Mathematical Society, 39:399–409, 1936. This is the paper in which the Pitman–Koopman theorem, sometimes called the Pitman–Koopman–Darmois theorem, appeared.
- "The axioms and algebra of intuitive probability", Annals of MathematicsAnnals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
41, 269–292, 1940. - "The bases of probability", Bulletin of the American Mathematical Society, 46, 763–774, 1940.
- "Intuitive probabilities and sequences", Annals of Mathematics 42, 169–187, 1941.
- Search and Screening, first edition 1946.