David Blackwell Instance for Activities That Could Run Sometime Between Now and the End of June 2012
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An Analog of the Minimax Theorem for Vector Payoffs
Pacific Journal of Mathematics AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL Vol. 6, No. 1 November 1956 AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL 1. Introduction* The von Neumann minimax theorem [2] for finite games asserts that for every rxs matrix M=\\m(i, j)\\ with real elements there exist a number v and vectors P=(Pi, •••, Pr)f Q={QU •••> Qs)f Pi, Qj>β, such that i> 3) for all i, j. Thus in the (two-person, zero-sum) game with matrix Λf, player I has a strategy insuring an expected gain of at least v, and player II has a strategy insuring an expected loss of at most v. An alternative statement, which follows from the von Neumann theorem and an appropriate law of large numbers is that, for any ε>0, I can, in a long series of plays of the game with matrix M, guarantee, with probability approaching 1 as the number of plays becomes infinite, that his average actual gain per play exceeds v — ε and that II can similarly restrict his average actual loss to v-he. These facts are assertions about the extent to which each player can control the center of gravity of the actual payoffs in a long series of plays. In this paper we investigate the extent to which this center of gravity can be controlled by the players for the case of matrices M whose elements m(i9 j) are points of ΛΓ-space. Roughly, we seek to answer the following question. -
Blackwell, David; Diaconis, Persi a Non-Measurable Tail Set
Blackwell, David; Diaconis, Persi A non-measurable tail set. Statistics, probability and game theory, 1--5, IMS Lecture Notes Monogr. Ser., 30, Inst. Math. Statist., Hayward, CA, 1996. Blackwell, David Operator solution of infinite $G\sb \delta$ games of imperfect information. Probability, statistics, and mathematics, 83--87, Academic Press, Boston, MA, 1989. 60040 Blackwell, David; Mauldin, R. Daniel Ulam's redistribution of energy problem: collision transformations. Lett. Math. Phys. 10 (1985), no. 2-3, 149--153. Blackwell, David; Dubins, Lester E. An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 (1983), no. 4, 691--692. 60B10 Blackwell, David; Maitra, Ashok Factorization of probability measures and absolutely measurable sets. Proc. Amer. Math. Soc. 92 (1984), no. 2, 251--254. Blackwell, David; Girshick, M. A. Theory of games and statistical decisions. Reprint of the 1954 edition. Dover Publications, Inc., New York, 1979. xi+355 pp. ISBN: 0-486-63831-6 90D35 (62Cxx) Blackwell, David On stationary policies. With discussion. J. Roy. Statist. Soc. Ser. 133 (1970), no. 1, 33--37. 90C40 Blackwell, David The stochastic processes of Borel gambling and dynamic programming. Ann. Statist. 4 (1976), no. 2, 370--374. Blackwell, David; Dubins, Lester E. On existence and non-existence of proper, regular, conditional distributions. Ann. Probability 3 (1975), no. 5, 741--752. Blackwell, David; MacQueen, James B. Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 (1973), 353--355. Blackwell, David; Freedman, David On the amount of variance needed to escape from a strip. Ann. Probability 1 (1973), 772--787. Blackwell, David Discreteness of Ferguson selections. -
Rational Inattention with Sequential Information Sampling∗
Rational Inattention with Sequential Information Sampling∗ Benjamin Hébert y Michael Woodford z Stanford University Columbia University July 1, 2016 Preliminary and Incomplete. WARNING: Some results in this draft are not corrected as stated. Please contact the authors before citing. Comments welcome. Abstract We generalize the rationalize inattention framework proposed by Sims [2010] to allow for cost functions other than Shannon’s mutual information. We study a prob- lem in which a large number of successive samples of information about the decision situation, each only minimally informative by itself, can be taken before a decision must be made. We assume that the cost required for each individual increment of in- formation satisfies standard assumptions about continuity, differentiability, convexity, and monotonicity with respect to the Blackwell ordering of experiments, but need not correspond to mutual information. We show that any information cost function sat- isfying these axioms exhibits an invariance property which allows us to approximate it by a particular form of Taylor expansion, which is accurate whenever the signals acquired by an agent are not very informative. We give particular attention to the case in which the cost function for an individual increment to information satisfies the ad- ditional condition of “prior invariance,” so that it depends only on the way in which the probabilities of different observations depend on the state of the world, and not on the prior probabilities of different states. This case leads to a cost function for the overall quantity of information gathered that is not prior-invariant, but “sequentially prior-invariant.” We offer a complete characterization of the family of sequentially prior-invariant cost functions in the continuous-time limit of our dynamic model of in- formation sampling, and show how the resulting theory of rationally inattentive choice differs from both static and dynamic versions of a theory based on mutual information. -
An Annotated Bibliography on the Foundations of Statistical Inference
• AN ANNOTATED BIBLIOGRAPHY ON THE FOUNDATIONS OF STATISTICAL INFERENCE • by Stephen L. George A cooperative effort between the Department of Experimental • Statistics and the Southeastern Forest Experiment Station Forest Service U.S.D.A. Institute of Statistics Mimeograph Series No. 572 March 1968 The Foundations of Statistical Inference--A Bibliography During the past two hundred years there have been many differences of opinion on the validity of certain statistical methods and no evidence that ,. there will be any general agreement in the near future. Also, despite attempts at classification of particular approaches, there appears to be a spectrum of ideas rather than the existence of any clear-cut "schools of thought. " The following bibliography is concerned with the continuing discussion in the statistical literature on what may be loosely termed ''the foundations of statistical inference." A major emphasis is placed on the more recent works in this area and in particular on recent developments in Bayesian analysis. Invariably, a discussion on the foundations of statistical inference leads one to the more general area of scientific inference and eventually to the much more general question of inductive inference. Since this bibliography is intended mainly for those statisticians interested in the philosophical foundations of their chosen field, and not for practicing philosophers, the more general discussion of inductive inference was deliberately de-emphasized with the exception of several distinctive works of particular relevance to the statistical problem. Throughout, the temptation to gather papers in the sense of a collector was resisted and most of the papers listed are of immediate relevance to the problem at hand. -
David Blackwell, 1919–2010: an Explorer In
RETROSPECTIVE David Blackwell, 1919–2010: An explorer in mathematics and statistics RETROSPECTIVE Peter J. Bickela,1 David Blackwell, a pioneering explorer who made foundational contributions to several branches of mathematics and statistics, passed away on July 8, 2010. He was born in Centralia, Illinois, on April 24, 1919, and, as his mathematical talents were recog- nized early, he entered the University of Illinois at Urbana–Champaign at age 16. Although racial dis- crimination affected his life and career in painful ways, his accomplishments were eventually rewarded with the honors they deserved, including election to the National Academy of Sciences in 1965 as the first Black member and the American Academy of Arts and Sciences in 1968. Nevertheless, his love of math- ematics, science, people, and his sunny personality prevailed. Blackwell left contributions that bear his name and other major ideas in five quite different areas of mathematics, statistics, and operations research. With limited opportunities available to him, Black- well initially thought of becoming an elementary school teacher. However, his professors at the Univer- sity of Illinois soon recognized his talent for mathe- matics and encouraged him to pursue graduate studies in the Illinois Mathematics program instead. During his graduate studies, Blackwell worked with Joseph Doob, one of the founding figures of modern David Blackwell. Image credit: The Blackwell family. probability theory, a National Academy of Sciences member, and National Medal of Science winner. In sequential analysis, game theory, and decision the- 1941, at age 22, he completed a doctoral thesis in the ory. They included: what is now known as the Rao- theory of Markov chains, a set of ideas to which he Blackwell theorem, a fundamental improvement frequently returned in his later work. -
The First One Hundred Years
The Maryland‐District of Columbia‐Virginia Section of the Mathematical Association of America: The First One Hundred Years Caren Diefenderfer Betty Mayfield Jon Scott November 2016 v. 1.3 The Beginnings Jon Scott, Montgomery College The Maryland‐District of Columbia‐Virginia Section of the Mathematical Association of America (MAA) was established, just one year after the MAA itself, on December 29, 1916 at the Second Annual Meeting of the Association held at Columbia University in New York City. In the minutes of the Council Meeting, we find the following: A section of the Association was established for Maryland and the District of Columbia, with the possible inclusion of Virginia. Professor Abraham Cohen, of Johns Hopkins University, is the secretary. We also find, in “Notes on the Annual Meeting of the Association” published in the February, 1917 Monthly, The Maryland Section has just been organized and was admitted by the council at the New York meeting. Hearty cooperation and much enthusiasm were reported in connection with this section. The phrase “with the possible inclusion of Virginia” is curious, as members from all three jurisdictions were present at the New York meeting: seven from Maryland, one from DC, and three from Virginia. However, the report, “Organization of the Maryland‐Virginia‐District of Columbia Section of the Association” (note the order!) begins As a result of preliminary correspondence, a group of Maryland mathematicians held a meeting in New York at the time of the December meeting of the Association and presented a petition to the Council for authority to organize a section of the Association in Maryland, Virginia, and the District of Columbia. -
Bayesian Statistics: Thomas Bayes to David Blackwell
Bayesian Statistics: Thomas Bayes to David Blackwell Kathryn Chaloner Department of Biostatistics Department of Statistics & Actuarial Science University of Iowa [email protected], or [email protected] Field of Dreams, Arizona State University, Phoenix AZ November 2013 Probability 1 What is the probability of \heads" on a toss of a fair coin? 2 What is the probability of \six" upermost on a roll of a fair die? 3 What is the probability that the 100th digit after the decimal point, of the decimal expression of π equals 3? 4 What is the probability that Rome, Italy, is North of Washington DC USA? 5 What is the probability that the sun rises tomorrow? (Laplace) 1 1 1 My answers: (1) 2 (2) 6 (3) 10 (4) 0.99 Laplace's answer to (5) 0:9999995 Interpretations of Probability There are several interpretations of probability. The interpretation leads to methods for inferences under uncertainty. Here are the 2 most common interpretations: 1 as a long run frequency (often the only interpretation in an introductory statistics course) 2 as a subjective degree of belief You cannot put a long run frequency on an event that cannot be repeated. 1 The 100th digit of π is or is not 3. The 100th digit is constant no matter how often you calculate it. 2 Similarly, Rome is North or South of Washington DC. The Mathematical Concept of Probability First the Sample Space Probability theory is derived from a set of rules and definitions. Define a sample space S, and A a set of subsets of S (events) with specific properties. -
David Haussler Elected to NAS IMS Member David Haussler Is One of 72 New Mem- Contents Bers Elected to the US National Academy of Sciences
Volume 35 Issue 5 IMS Bulletin June 2006 David Haussler elected to NAS IMS member David Haussler is one of 72 new mem- CONTENTS bers elected to the US National Academy of Sciences. 1 David Haussler, NAS Members are chosen in recognition of their distin- guished and continuing achievements in original 2 IMS Members’ News: Bin Yu, Hongyu Zhao, research. Ralph Cicerone, Academy president, says, Jianqing Fan “Election to the Academy is considered one of the highest honors in American science and engineering.” 4 LNMS Sale David Haussler is an investigator with the 5 IMS news Howard Hughes Medical Institute and a professor 6 Rio guide: eating, dancing, of biomolecular engineering at the University of having fun California, Santa Cruz, where he directs the Center David Haussler, recently elected to the for Biomolecular Science & Engineering. He serves 7 SSP report US National Academy of Sciences as scientific co-director for the California Institute 8 Terence’s Stuff: The Big for Quantitative Biomedical Research (QB3), and he is a consulting professor at Question both Stanford Medical School and UC San Francisco Biopharmaceutical Sciences 9 Meet the Members Department. David was recently elected to the American Academy of Arts and Sciences; 10 JSM News: IMS Invited Program highlights he is also a fellow of both AAAS and AAAI. He has won a number of prestigious awards, including recently the 2006 Dickson Prize for Science from Carnegie Mellon IMS Meetings 12 University. David received his BA degree in mathematics from Connecticut College 19 Other Meetings and in 1975, an MS in applied mathematics from California Polytechnic State University Announcements at San Luis Obispo in 1979, and his PhD in computer science from the University of 21 Employment Colorado at Boulder in 1982. -
IMS Bulletin 38(7)
Volume 38 • Issue 7 IMS Bulletin August/September 2009 Election results announced CONTENTS The results are announced for the 2009 1 Election results IMS Elections. The next President-Elect will be Peter Hall. Peter is Professor 2 Members’ News: Jean Opsomer; Len Stefanski; of Statistics in the Department of Klaus Krickeberg Mathematics and Statistics, University of Melbourne; and Professor of Statistics, 3 Nominations sought Department of Statistics, UC Davis (a Peter Hall 4 Medallion preview: Gábor fractional appointment). He will serve Lugosi on the IMS Executive Committee for three years, as President- 5 Members’ Discoveries: Elect (2009–10), President (2010–2011) and Past President Extending the scope of (2011–2012). Empirical Likelihood Five Council members have been elected for a three-year term, 6 Meeting report: WNAR to serve from August 2009 to August 2012. In alphabetical order, they are: 8 Probability and statistics in Marie Davidian Africa (William Neal Reynolds Professor of Statistics in the Department of Statistics at North Carolina State University) 9 National Academies Edward George (Universal Furniture Professor and Chairman induction ceremony of the Department of Statistics, Wharton School, University of 11 New CUP IMS Monographs Pennsylvania) and IMS Textbooks series Robert Tibshirani (Professor in the Departments of Health 12 Obituary: Kai Lai Chung Research & Policy and Statistics, Stanford University) Michael Titterington (Professor in the Department of Statistics 13 Niss.org: updated at the University of Glasgow, UK), and 15 Rick’s Ramblings: Coming Zhiliang Ying (Professor in the Department of Statistics at soon: The Fourth Edition Columbia University). 16 Terence’s Stuff: “The Flu In addition, this year another Council member has been and I” elected for a two-year term, to finish an unexpired term on a Davar Khoshnevisan 17 IMS meetings vacated Council position. -
September 1988 Table of Contents
OTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1988 Steele Prizes page 965 ;I~ The AMS Centennial: Social and Mathematical Festivities page 970 SEPTEMBER 1988, VOLUME 35, NUMBER 7 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences This calendar lists all meetings which have been approved prior to Mathematical Society in the issue corresponding to that of the Notices the date this issue of Notices was sent to the press. The summer which contains the program of the meeting. Abstracts should be sub and annual meetings are joint meetings of the Mathematical Associ mitted on special forms which are available in many departments of ation of America and the American Mathematical Society. The meet mathematics and from the headquarters office of the Society. Ab ing dates which fall rather far in the future are subject to change; this stracts of papers to be presented at the meeting must be received is particularly true of meetings to which no numbers have been as at the headquarters of the Society in Providence, Rhode Island, on signed. Programs of the meetings will appear in the issues indicated or before the deadline given below for the meeting. Note that the below. First and supplementary announcements of the meetings will deadline for abstracts for consideration for presentation at special have appeared in earlier issues. sessions is usually three weeks earlier than that specified below. For Abstracts of papers presented at a meeting of the Society are pub additional information, consult the meeting announcements and the lished in the journal Abstracts of papers presented to the American list of organizers of special sessions. -
15/14/39 Liberal Arts and Sciences Mathematics Department Donald L. Burkholder Papers, 1947-2011 Box 1
15/14/39 Liberal Arts and Sciences Mathematics Department Donald L. Burkholder Papers, 1947-2011 Box 1: Biographical File CORRESPONDENCE--This correspondence file contains confidential information, and access requires approval from an archivist. Family, 1961-92 Politics (includes letters from Senators Paul Douglas and Everett McKinley Dirksen regarding civil rights legislation), 1963-64 Professional, 1956-75 1959-65 1962-64 1967 1968-71 1969-70 1973-76 1981-86 1987-91 1992-94 1994-2003 2004-2011 By Correspondent David Aldous, 1979-81 A. N. Al-Hussaini, 1964-2003 Dale Alspach, 1985-89 Madjid Amir, 1989-94 F. Amoroso, 1996 George Anastassiou, 1991 T. W. Anderson, 1965 T. Ando, 1983 J. Arazy, 1978-91 Ludwig Arnold, 1993-94 J. Marshall Ash, 1986-87 Nakhle Asmar, 1992-97 Idris Assani, 1986-95 Robert J. Aumann, 1986 Don G. Austin, 1966 Al Baernstein, 1975-2003 R. R. Bahadur, 1997 Rodrigo Banuelos, 1985-2005 (4 folders) 15/14/39 2 Box 2: 1985-2005 (4 folders) Martin Barlow, 1990-95 (2 folders) Richard Bass, 1985-2007 (2 folders) J. G. Basterfield, 1972 Robert Bauer, 1994-2003 (3 folders) Bernard Beauzamy, 1986-2008 (5 folders) William Beckner, 1980-88 Denis Bell, 1995-97 Alexandra Bellow, 1965-2001 Box 3: 1965-2001 John Benedetto, 1996 Andrew Bennett, 1986-87 Colin Bennett, 1994 Robert Berk, 1967-68 Klaus Bichteler, 1981-92 P. Billard, 1972 N. H. Bingham, 1974-93 Oscar Blasco, 1986-90 J. R. Blum, 1954-58 R. M. Blumenthal, 1959-68 Colin R. Blyth, 1954-2003 Robert Bohrer, 1971-94 Bela Bollobas, 1979-96 Christer Borell, 1974-96 Jean Bourgain, 1982-2006 (8 folders) Nadjis Bouzar, 1989 J. -
Notices: Highlights
Salt Lake City Meeting (August 5-8)- Page 761 Notices of the American Mathematical Society August 1987, Issue 257 Volume 34, Number 5, Pages 729-872 Providence, Rhode Island USA ISSN 0002-9920 Calendar of AMS Meetings , THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of Notices was sent to the press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change: this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and supplementary announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meeting. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the headquarter's office of the Society. Abstracts of papers to be presented at the meeting must be recejvedat the headquarters of the Society in Providence. Rhode Island. on or before the deadline given below for the meeting. Note that the deadline for abstracts for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information. consult the meeting announcements and the list of organizers of special sessions.