Herbert Robbins
Encyclopedia
Herbert Ellis Robbins was an American mathematician
and statistician
who did research in topology
, measure theory, statistics
, and a variety of other fields. He was the co-author, with Richard Courant
, of What is Mathematics?
, a popularization that is still in print. The Robbins lemma
, used in empirical Bayes method
s, is named after him. Robbins algebra
s are named after him because of a conjecture (since proved) that he posed concerning Boolean algebras. The Robbins theorem
, in graph theory
, is also named after him, as is the Whitney–Robbins synthesis
, a tool he introduced to prove this theorem. The well-known unsolved problem of minimizing in sequential selection the expected rank of the selected item under full information, sometimes referred to as the fourth secretary problem
, also bears his name: Robbins' problem (of optimal stopping)
.
, Pennsylvania
.
As an undergraduate, Robbins attended Harvard University
, where Marston Morse
influenced him to become interested in mathematics. Robbins received a doctorate
from Harvard in 1938 under the supervision of Hassler Whitney
and was an instructor at New York University
from 1939 to 1941. After World War II
, Robbins taught at the University of North Carolina at Chapel Hill
from 1946 to 1952, where he was one of the original members of the department of mathematical statistics, then spent a year at the Institute for Advanced Study
. In 1953, he became a professor of mathematical statistics at Columbia University
. He retired from full-time activity at Columbia in 1985 and was then a professor at Rutgers University
until his retirement in 1997.
In 1955, Robbins introduced empirical Bayes method
s at the Third Berkeley Symposium on Mathematical Statistics and Probability. Robbins was also one of the inventors of the first stochastic approximation
algorithm, the Robbins-Monro method, and worked on the theory of power-one tests and optimal stopping
.
He died in 2001, in Princeton
, New Jersey
.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and statistician
Statistician
A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...
who did research in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, measure theory, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, and a variety of other fields. He was the co-author, with Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
, of What is Mathematics?
What is Mathematics?
What Is Mathematics? is a mathematics book written by Richard Courant and Herbert Robbins, published in England by Oxford University Press. It is an introduction to mathematics, intended both for the mathematics student and for the general public....
, a popularization that is still in print. The Robbins lemma
Robbins lemma
In statistics, the Robbins lemma, named after Herbert Robbins, states that if X is a random variable with a Poisson distribution, and f is any function for which the expected value E exists, then...
, used in empirical Bayes method
Empirical Bayes method
Empirical Bayes methods are procedures for statistical inference in which the prior distribution is estimated from the data. This approach stands in contrast to standardBayesian methods, for which the prior distribution is fixed before any data are observed...
s, is named after him. Robbins algebra
Robbins algebra
In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by \lor, and a single unary operation usually denoted by \neg...
s are named after him because of a conjecture (since proved) that he posed concerning Boolean algebras. The Robbins theorem
Robbins theorem
In graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge that makes it into a strongly connected graph. Robbins' theorem, named after , states that the graphs that have strong orientations are exactly the 2-edge-connected graphs...
, in graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, is also named after him, as is the Whitney–Robbins synthesis
Ear decomposition
In graph theory, an ear of an undirected graph G is a path P where the two endpoints of the path may coincide, but where otherwise no repetition of edges or vertices is allowed, so every internal vertex of P has degree two in P...
, a tool he introduced to prove this theorem. The well-known unsolved problem of minimizing in sequential selection the expected rank of the selected item under full information, sometimes referred to as the fourth secretary problem
Secretary problem
The secretary problem is one of many names for a famous problem of theoptimal stopping theory.The problem has been studied extensively in the fields ofapplied probability, statistics, and decision theory...
, also bears his name: Robbins' problem (of optimal stopping)
Robbins' problem (of optimal stopping)
In probability theory, Robbins' problem of optimal stopping, named after Herbert Robbins, is sometimes referred to as the fourth secretary problem or the problem of minimizing the expected rank with full information. Its statement is as follows....
.
Biography
Robbins was born in New CastleNew Castle, Pennsylvania
New Castle is a city in Lawrence County, Pennsylvania, United States, northwest of Pittsburgh and near the Pennsylvania-Ohio border just east of Youngstown, Ohio; in 1910, the total population was 36,280; in 1920, 44,938; and in 1940, 47,638. The population has fallen to 26,309 according to the...
, Pennsylvania
Pennsylvania
The Commonwealth of Pennsylvania is a U.S. state that is located in the Northeastern and Mid-Atlantic regions of the United States. The state borders Delaware and Maryland to the south, West Virginia to the southwest, Ohio to the west, New York and Ontario, Canada, to the north, and New Jersey to...
.
As an undergraduate, Robbins attended Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, where Marston Morse
Marston Morse
Harold Calvin Marston Morse was an American mathematician best known for his work on the calculus of variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory...
influenced him to become interested in mathematics. Robbins received a doctorate
Doctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...
from Harvard in 1938 under the supervision of Hassler Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
and was an instructor at New York University
New York University
New York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
from 1939 to 1941. After World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, Robbins taught at the University of North Carolina at Chapel Hill
University of North Carolina at Chapel Hill
The University of North Carolina at Chapel Hill is a public research university located in Chapel Hill, North Carolina, United States...
from 1946 to 1952, where he was one of the original members of the department of mathematical statistics, then spent a year at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
. In 1953, he became a professor of mathematical statistics at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
. He retired from full-time activity at Columbia in 1985 and was then a professor at Rutgers University
Rutgers University
Rutgers, The State University of New Jersey , is the largest institution for higher education in New Jersey, United States. It was originally chartered as Queen's College in 1766. It is the eighth-oldest college in the United States and one of the nine Colonial colleges founded before the American...
until his retirement in 1997.
In 1955, Robbins introduced empirical Bayes method
Empirical Bayes method
Empirical Bayes methods are procedures for statistical inference in which the prior distribution is estimated from the data. This approach stands in contrast to standardBayesian methods, for which the prior distribution is fixed before any data are observed...
s at the Third Berkeley Symposium on Mathematical Statistics and Probability. Robbins was also one of the inventors of the first stochastic approximation
Stochastic approximation
Stochastic approximation methods are a family of iterative stochastic optimization algorithms that attempt to find zeroes or extrema of functions which cannot be computed directly, but only estimated via noisy observations....
algorithm, the Robbins-Monro method, and worked on the theory of power-one tests and optimal stopping
Optimal stopping
In mathematics, the theory of optimal stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance...
.
He died in 2001, in Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, New Jersey
New Jersey
New Jersey is a state in the Northeastern and Middle Atlantic regions of the United States. , its population was 8,791,894. It is bordered on the north and east by the state of New York, on the southeast and south by the Atlantic Ocean, on the west by Pennsylvania and on the southwest by Delaware...
.
Selected writings
- A theorem on graphs with an application to a problem on traffic control, American Mathematical Monthly, vol. 46 (1939), pp. 281-283.
- What is Mathematics? An Elementary Approach to Ideas and Methods, with Richard Courant, London: Oxford University Press, 1941.
- The central limit theoremCentral limit theoremIn probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
for dependent random variableRandom variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s, with Wassily HoeffdingWassily HoeffdingWassily Hoeffding was an American statistician and probabilist...
, Duke Mathematical Journal, vol. 15 (1948), pp. 773–780. - A stochastic approximationStochastic approximationStochastic approximation methods are a family of iterative stochastic optimization algorithms that attempt to find zeroes or extrema of functions which cannot be computed directly, but only estimated via noisy observations....
method, with Sutton Monro, Annals of Mathematical Statistics, vol. 22, no. 3 (September 1951), pp. 400–407. - An empirical Bayes approach to statistics, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Jerzy Neyman, ed., vol. 1, Berkeley, California: University of California Press, 1956, pp. 157–163.