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31:
435:— that bear little outward resemblance to the original definition, but each of which satisfies our intuitive notion of properties that random sequences ought to have: random sequences should be incompressible, they should pass statistical tests for randomness, and it should be impossible to make money
705:
In the 1970s, Per Martin-Löf made important contributions to statistical theory and inspired further research, especially by
Scandinavian statisticians including Rolf Sundberg, Thomas Höglund, and Steffan Lauritzen. In this work, Martin-Löf's previous research on probability measures on semigroups
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on them. The existence of these multiple definitions of Martin-Löf randomness, and the stability of these definitions under different models of computation, give evidence that Martin-Löf randomness is a fundamental property of mathematics and not an accident of Martin-Löf's particular model. The
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718:, using finite-sample principles. Before (and after) Martin-Löf, such nested models have often been tested using chi-square hypothesis tests, whose justifications are only asymptotic (and so irrelevant to real problems, which always have finite samples).
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is now emeritus professor of mathematical statistics at
Stockholm University; the two brothers have collaborated in research in probability and statistics. The research of Anders and Per Martin-Löf has influenced statistical theory, especially concerning
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sequence of characters, all of whose prefixes (except possibly a finite number of exceptions) are strings that are "close to" algorithmically random (their length is within a constant of their
Kolmogorov complexity).
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properties of infinite sequences failed to provide a foundation for statistics, which considers only finite samples. Much of Martin-Löf's work in statistics was to provide a finite-sample foundation for statistics.
1851:
Sundberg, Rolf Some results about decomposable (or Markov-type) models for multidimensional contingency tables: distribution of marginals and partitioning of tests. Scand. J. Statist. 2 (1975), no. 2, 71—79.
1752:
Martin-Löf, P. The notion of redundancy and its use as a quantitative measure of the deviation between a statistical hypothesis and a set of observational data. With a discussion by F. Abildgård,
1955:
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in order to define a random sequence as one that passed all tests for randomness; however, the precise notion of a randomness test was left vague. Martin-Löf's key insight was to use the
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Sundberg, Rolf An iterative method for solution of the likelihood equations for incomplete data from exponential families. Comm. Statist.—Simulation Comput. B5 (1976), no. 1, 55—64.
1709:
Statistika
Modeller (Statistical Models): Anteckningar fran seminarier läsåret 1969–1970 (Notes from seminars in the academic year 1969–1970), with the assistance of Rolf Sundberg.
1465:
Statistika
Modeller (Statistical Models): Anteckningar fran seminarier läsåret 1969–1970 (Notes from seminars in the academic year 1969–1970), with the assistance of Rolf Sundberg.
1697:. 1963. "Utvärdering av livslängder i subnanosekundsområdet" ("Evaluation of lifetimes in time-lengths below one nanosecond"). ("Sundberg formula", according to Sundberg 1971)
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which was discovered by Girard when studying System U, an inconsistent extension of System F. This experience led Per Martin-Löf to develop the philosophical foundations of
1895:
Per Martin-Löf. An intuitionistic theory of types: Predicative part. In H. E. Rose and J. C. Shepherdson, editors, Logic
Colloquium ‘73, pages 73–118. North Holland, 1975.
1964:
1892:
In G. Sambin and J. Smith, editors, Twenty-Five Years of
Constructive Type Theory. Oxford University Press, 1998. Reprinted version of an unpublished report from 1972.
1737:
Martin-Löf, P. Repetitive structures and the relation between canonical and microcanonical distributions in statistics and statistical mechanics. With a discussion by
2010:
1783:
Martin-Löf, Per The notion of redundancy and its use as a quantitative measure of the discrepancy between a statistical hypothesis and a set of observational data.
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Philosophy and
Foundations of Mathematics: Epistemological and Ontological Aspects. A conference dedicated to Per Martin-Löf on the occasion of his retirement
1972:
Proof-Theoretic
Semantics: Assessment and Future Perspectives. Proceedings of the Third Tübingen Conference on Proof-Theoretic Semantics, 27–30 March 2019
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is at least the length of the string. This is a different meaning from the usage of the term in statistics. Whereas statistical randomness refers to the
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1441:. 1963. "Utvärdering av livslängder i subnanosekundsområdet" ("Evaluation of lifetimes in time-lengths below one nanosecond"). ("Sundberg formula")
2060:
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type theory as presented in his 1984 Bibliopolis book, and extended in a number of increasingly philosophical texts, such as his influential
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thesis that the definition of Martin-Löf randomness "correctly" captures the intuitive notion of randomness has been called the "Martin-Löf–
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2050:
1704:. Lecture notes, Mathematical Institute, Aarhus University. ("Sundberg formula" credited to Anders Martin-Löf, according to Sundberg 1971)
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that produces the string (e.g. flipping a coin to produce each bit will randomly produce a string), algorithmic randomness refers to the
2040:
2035:
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559:, in a Swedish zoological journal: This paper was soon cited in leading international journals, and this paper continues to be cited.
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Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable
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Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable
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744:. Sundberg credits a formula, later known as the Sundberg formula, to previous manuscripts of the Martin-Löf brothers, Per and
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in 1990, which was heavily influenced by his later ideas, intensional, and more amenable to being implemented on a computer.
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Martin-Löf wrote a licenciate thesis on probability on algebraic structures, particularly semigroups, while a student of
1564:
Arkiv för
Zoologi (Zoology Files), Kungliga Svenska Vetenskapsakademien (The Royal Swedish Academy of Sciences) Serie 2
471:. Algorithmic information theory separates random from nonrandom strings in a way that is relatively invariant to the
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1749:(Aarhus, 1973), pp. 271–294. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974.
1734:(Aarhus, 1973), pp. 121–138. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974.
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defines a random string as one that cannot be produced from any computer program that is shorter than the string (
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2020:
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led to a notion of "repetitive structure" and a novel treatment of sufficient statistics, in which one-parameter
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in his youth and remains an enthusiastic bird-watcher. As a teenager, he published an article on estimating the
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1780:(Aarhus, 1973), pp. 1–42. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974.
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1454:. Lecture notes, Mathematical Institute, Aarhus University. ("Sundberg formula" credited to Anders Martin-Löf).
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1995:
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Martin-Löf, P. Exact tests, confidence regions and estimates. Foundations of probability and statistics. II.
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of a bird, which is extremely difficult for humans, is one of the first examples in Martin-Löf's lectures on
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Martin-Löf, P. (1961). "Mortality rate calculations on ringed birds with special reference to the Dunlin
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1919:(Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984.
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to define formally the notion of a test for randomness. This contrasts with the idea of randomness in
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748:. Many of these results reached the international scientific community through the 1976 paper on the
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Martin-Löf randomness has since been shown to admit many equivalent characterizations — in terms of
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1971:
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Until his retirement in 2009, Per Martin-Löf held a joint chair for Mathematics and Philosophy at
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The 1984 type theory was extensional while the type theory presented in the book by Nordström
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as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced
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Does HoTT Provide a Foundation for Mathematics? by James Ladyman (University of Bristol, UK)
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Sundberg, Rolf. Maximum likelihood theory for incomplete data from an exponential family.
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578:. Martin-Löf's first paper discussed the problem of estimating the mortality rates of the
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8:
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953:. A number of popular computer-based proof systems are based on type theory, for example
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S. M. Taylor (1966). "Recent Quantitative Work on British Bird Populations. A Review".
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Martin-Löf, P. "Exact tests, confidence regions and estimates", with a discussion by
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1173:. Swedish Collegium for Advanced Study, Uppsala, May 5–8, 2009. Retrieved 2014-01-26.
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424:; in that theory, no particular element of a sample space can be said to be random.
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On the Meanings of the Logical Constants and the Justifications of the Logical Laws
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On the Meanings of the Logical Constants and the Justifications of the Logical Laws
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1977:
1865:. Lecture Notes in Statistics, 49. Springer-Verlag, New York, 1988. xvi+268 pp.
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1991:
1838:. Dissertation, Institute for Mathematical Statistics, Stockholm University.
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Proceedings of Conference on Foundational Questions in Statistical Inference
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Proceedings of Conference on Foundational Questions in Statistical Inference
1732:
Proceedings of Conference on Foundational Questions in Statistical Inference
1429:. Dissertation, Institute for Mathematical Statistics, Stockholm University.
892:. Martin-Löf's first draft article on type theory dates back to 1971. This
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808:
764:, which was published in a leading international journal, sponsored by the
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Höglund, Thomas. The exact estimate — a method of statistical estimation.
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Martin-Löf's student, Rolf Sundberg, developed a detailed analysis of the
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1882:
Per Martin-Löf. A theory of types. Preprint, Stockholm University, 1971.
1488:(1977). "Maximum Likelihood from Incomplete Data via the EM Algorithm".
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philosophical traditions, especially of the German-language writings of
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In 1964 and 1965, Martin-Löf studied in Moscow under the supervision of
1503:
1348:
1078:
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280:. Since the late 1970s, Martin-Löf's publications have been mainly in
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1909:. Eds. Cohen, et al. North-Holland, Amsterdam. pp. 153–175, 1982.
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1738:
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for estimation using data from exponential families, especially with
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57:
1340:
1110:, Europa Publications, 1996, p. 1020: "Martin-Löf, Per Erik Rutger."
272:. He is internationally renowned for his work on the foundations of
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432:
261:
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Martin-Löf, P. The continuity theorem on a locally compact group.
1361:
Martin-Löf, P. The continuity theorem on a locally compact group.
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1940:. Oxford University Press, 1990. (The book is out of print, but
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1723:
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369:; his first scientific publication was on the mortality rates of
30:
941:
Martin-Löf's intuitionistic type theory developed the notion of
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520:
258:
1797:
Martin-Löf, Per Reply to Erling Sverdrup's polemical article:
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that gave the first suitable definition of a random sequence.
1640:
An Introduction to Kolmogorov Complexity and Its Applications
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954:
281:
175:
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Martin-Löf, Per. Probability theory on discrete semigroups.
1651:
Probability on algebraic structures, following Ulf Grenander
880:
From 1968 to '69 he worked as an assistant professor at the
16:
Swedish logician, philosopher, and mathematical statistician
1374:
Martin-Löf, Per Probability theory on discrete semigroups.
1131:
Peter Dybjer on types and testing – The Type Theory Podcast
1702:
Statistics from the point of view of statistical mechanics
1452:
Statistics from the point of view of statistical mechanics
675:
The steps of the EM algorithm on a two component Gaussian
237:
1680:
Nitis Mukhopadhyay. "A Conversation with Ulf Grenander".
1589:
The Estimation of Animal Abundance and Related Parameters
1146:
Programming in Martin-Löf 's Type Theory: An Introduction
1143:
Nordström, Bengt; Petersson, Kent; Smith, Jan M. (1990),
666:
1631:
Per Martin-Löf. "The Definition of Random Sequences."
722:
Expectation–maximization method for exponential families
1877:
Foundations of mathematics, logic, and computer science
1387:
Nitis Mukhopadhyay. A Conversation with Ulf Grenander.
1142:
821:, Per Martin-Löf has published papers on the theory of
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1863:
Extremal families and systems of sufficient statistics
597:
510:
1907:
Logic, Methodology and Philosophy of Science VI, 1979
240:
234:
1936:Bengt Nordström, Kent Petersson, and Jan M. Smith.
1605:
1491:Journal of the Royal Statistical Society, Series B
1476:
1329:Journal of the Royal Statistical Society, Series D
1901:Constructive mathematics and computer programming
1551:
1291:Randomness, Unpredictability and Absence of Order
288:, Martin-Löf has wrestled with the philosophy of
2011:Members of the Royal Swedish Academy of Sciences
2002:
1969:In T. Piecha and P. Schroeder-Heister, editors,
1933:, 1992. Notes from a lecture given in Göteborg.
1856:Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
1675:Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
1376:Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
945:and directly influenced the development of the
1772:, O. Barndorff-Nielsen, J. D. Kalbfleisch and
1608:Hierarchical Modeling and Inference in Ecology
1258:(1966). "The definition of random sequences".
650:. In his paper "On Tables of Random Numbers",
646:Martin-Löf developed innovative approaches to
495:Per Martin-Löf has done important research in
888:with whom he discussed issues related to the
354:Per Martin-Löf received his PhD in 1970 from
1978:http://dx.doi.org/10.15496/publikation-35319
1326:
499:, which (in the Swedish tradition) includes
412:had attempted to formalize the notion of a
312:, Martin-Löf has been active in developing
1625:
1557:
1467:Stockholm University. ("Sundberg formula")
1403:
1399:
1397:
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29:
1924:Philosophical implications of type theory
1790:Sverdrup, Erling. "Tests without power."
1688:
1591:. Caldwel, New Jersey: Blackburn Press.
1248:
903:. However, this system turned out to be
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670:
586:methods. The problem of determining the
514:
1938:Programming in Martin-Löf's Type Theory
1583:, 4 December 1999, magazine issue 2215.
1532:. The Royal Swedish Academy of Sciences
1407:(1963). "On Tables of Random Numbers".
1394:
1317:, 4 December 1999, magazine issue 2215.
1202:
444:Thesis"; it is somewhat similar to the
2061:Academic staff of Stockholm University
2003:
1722:, D. A. Sprott, O. Barndorff-Nielsen,
1238:
1236:
786:
667:Model selection and hypothesis testing
276:, statistics, mathematical logic, and
1956:Nordic Journal of Philosophical Logic
1820:2 (1975), 161—165) by P. Martin-Löf.
1586:
631:
252:
1684:Volume 21, Number 3 (2006), 404–426.
1606:Royle, J. A.; R. M. Dorazio (2008).
1391:Volume 21, Number 3 (2006), 404–426.
1181:
1179:
1108:The International Who's Who: 1996-97
750:expectation–maximization (EM) method
738:expectation–maximization (EM) method
377:Randomness and Kolmogorov complexity
112:expectation–maximization (EM) method
2056:21st-century Swedish mathematicians
2051:20th-century Swedish mathematicians
1661:Probability on Algebraic Structures
1233:
598:Probability on algebraic structures
511:Bird-watching and sex determination
13:
1812:Sverdrup, Erling. A rejoinder to:
732:Expectation–maximization algorithm
710:were characterized. He provided a
403:The definition of random sequences
14:
2092:
2041:21st-century Swedish philosophers
2036:20th-century Swedish philosophers
1985:
1888:An intuitionistic theory of types
1220:The Mathematics Genealogy Project
1176:
987:Royal Swedish Academy of Sciences
829:, etc. He has been interested in
296:, partly inspired by the work of
133:Royal Swedish Academy of Sciences
574:, there are several problems of
230:
1518:
1470:
1457:
1444:
1432:
1419:
1381:
1368:
1355:
1320:
1300:
1280:
480:algorithmically random sequence
383:Algorithmically random sequence
341:expectation–maximization method
1981:. University of Tübingen 2019.
1668:Teor. Verojatnost. i Primenen.
1644:Introduction chapter full-text
1552:Bird watching and missing data
1363:Teor. Verojatnost. i Primenen.
1208:
1195:
1157:
1135:
1124:
1113:
1101:
985:Martin-Löf is a member of the
848:
453:algorithmic information theory
393:Algorithmic information theory
365:Martin-Löf is an enthusiastic
1:
2046:Swedish information theorists
1996:Mathematics Genealogy Project
1546:
1274:10.1016/S0019-9958(66)80018-9
457:Chaitin–Kolmogorov randomness
451:Following Martin-Löf's work,
2016:Members of Academia Europaea
1638:Li, Ming and Vitányi, Paul,
1297:, p. 145–167, Springer 1993.
408:Earlier researchers such as
7:
1805:2 (1975), no. 3, 158–160).
1776:and a reply by the author.
1745:and a reply by the author.
996:
890:Curry–Howard correspondence
10:
2097:
1926:, Unpublished notes, 1987?
1915:Intuitionistic type theory
1831:36 (1977), no. 2, 195—206.
989:(elected 1990) and of the
861:Intuitionistic type theory
858:
852:
813:Phenomenology (philosophy)
790:
725:
686:
635:
601:
555:of birds, using data from
528:
401:. He wrote a 1966 article
386:
380:
314:intuitionistic type theory
226:Per Erik Rutger Martin-Löf
1944:has been made available.)
1824:4 (1977), no. 3, 136—138.
1809:2 (1975), no. 3, 161–165.
1794:2 (1975), no. 3, 158–160.
1295:Philosophy of Probability
1152:, Oxford University Press
980:
947:calculus of constructions
921:proof-theoretic semantics
873:Martin-Löf has worked in
865:Proof-theoretic semantics
766:Royal Statistical Society
716:nested statistical models
628:at Stockholm University.
270:mathematical statistician
219:
207:
189:
167:
160:
125:
91:
81:
73:
65:
40:
28:
21:
1205:section of this article.
1095:
771:
604:probability distribution
431:, randomness tests, and
257:; born 8 May 1942) is a
1845:1 (1974), no. 2, 49—58.
1633:Information and Control
1626:Probability foundations
1261:Information and Control
1192:, retrieved 2014-01-26.
497:mathematical statistics
491:Mathematical statistics
459:); i.e. a string whose
180:Mathematical statistics
2071:Swedish ornithologists
2021:Mathematical logicians
1975:, pages 227-235. URI:
1861:Lauritzen, Steffen L.
1787:1 (1974), no. 1, 3—18.
1707:Per Martin-Löf. 1970.
1700:Per Martin-Löf. 1966.
1689:Statistics foundations
1635:, 9(6): 602–619, 1966.
1587:Seber, G.A.F. (2002).
1463:Per Martin-Löf. 1970.
1450:Per Martin-Löf. 1966.
1064:Charles Sanders Peirce
805:Continental philosophy
783:
684:
526:
254:[ˈmǎʈːɪnˈløːv]
116:Martin-Löf type theory
35:Per Martin-Löf in 2004
2031:Swedish statisticians
1931:Substitution calculus
1834:Rolf Sundberg. 1971.
1711:Stockholm University.
1425:Rolf Sundberg. 1971.
1405:Kolmogorov, Andrei N.
1201:For details, see the
882:University of Chicago
779:
674:
656:frequency probability
547:Per Martin-Löf began
518:
461:Kolmogorov complexity
418:theory of computation
389:Kolmogorov complexity
198:University of Chicago
120:Martin-Löf randomness
108:Sufficient statistics
1959:, 1(1): 11–60, 1996.
1039:Andrei N. Kolmogorov
951:logical framework LF
886:William Alvin Howard
708:exponential families
693:Sufficient statistic
473:model of computation
446:Church–Turing thesis
399:Andrei N. Kolmogorov
356:Stockholm University
337:exponential families
325:Stockholm University
214:Andrei N. Kolmogorov
194:Stockholm University
104:Repetitive structure
86:Stockholm University
1858:29 (1974), 257—271.
1814:Tests without power
1799:Tests without power
1577:"Gone Birdwatching"
1311:"Gone Birdwatching"
1203:#Statistical models
917:meaning explanation
896:theory generalized
823:logical consequence
819:philosophical logic
801:Analytic philosophy
793:Logical consequence
787:Philosophical logic
414:test for randomness
290:logical consequence
286:philosophical logic
1843:Scand. J. Statist.
1822:Scand. J. Statist.
1818:Scand. J. Statist.
1807:Scand. J. Statist.
1803:Scand. J. Statist.
1792:Scand. J. Statist.
1785:Scand. J. Statist.
1642:, Springer, 1997.
1287:Jean-Paul Delahaye
1169:2014-02-02 at the
923:, which justifies
877:for many decades.
875:mathematical logic
784:
754:Arthur P. Dempster
712:category-theoretic
685:
654:observed that the
648:statistical theory
638:Exponential family
632:Statistical models
592:statistical models
539:Mark and recapture
527:
501:probability theory
310:mathematical logic
2026:Swedish logicians
1912:Per Martin-Löf.
1695:Anders Martin-Löf
1663:. (Dover reprint)
1617:978-1-930665-55-2
1573:George A. Barnard
1439:Anders Martin-Löf
1307:George A. Barnard
1243:Martin-Löf (1961)
1190:Academia Europaea
1044:Anders Martin-Löf
991:Academia Europaea
642:Statistical model
616:Laplace transform
608:harmonic analysis
584:capture-recapture
524:(Calidris alpina)
410:Richard von Mises
360:Andrey Kolmogorov
332:Anders Martin-Löf
223:
222:
202:Aarhus University
162:Scientific career
149:Rolf Schock Prize
2088:
1965:Logic and Ethics
1962:Per Martin-Löf.
1947:Per Martin-Löf.
1929:Per Martin-Löf.
1922:Per Martin-Löf.
1898:Per Martin-Löf.
1885:Per Martin-Löf.
1768:, D. A. Sprott,
1766:A. W. F. Edwards
1716:A. W. F. Edwards
1670:10 1965 367–371.
1621:
1602:
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1526:"Per Martin-Løf"
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1365:10 1965 367–371.
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1216:"Per Martin-Löf"
1212:
1206:
1199:
1193:
1183:
1174:
1161:
1155:
1153:
1151:
1139:
1133:
1128:
1122:
1117:
1111:
1105:
1074:Bertrand Russell
1049:John von Neumann
993:(elected 1989).
909:Girard's paradox
831:Central-European
318:computer science
278:computer science
256:
251:
247:
246:
243:
242:
239:
236:
209:Doctoral advisor
172:Computer Science
153:
145:
137:
96:Random sequences
54:
50:
48:
33:
19:
18:
2096:
2095:
2091:
2090:
2089:
2087:
2086:
2085:
2001:
2000:
1988:
1879:
1691:
1653:
1628:
1618:
1599:
1566:. Band 13 (21).
1560:Calidris alpina
1554:
1549:
1544:
1535:
1533:
1524:
1523:
1519:
1475:
1471:
1462:
1458:
1449:
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1437:
1433:
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1402:
1395:
1386:
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1373:
1369:
1360:
1356:
1341:10.2307/2986734
1325:
1321:
1305:
1301:
1285:
1281:
1256:Martin-Löf, Per
1253:
1249:
1241:
1234:
1224:
1222:
1214:
1213:
1209:
1200:
1196:
1184:
1177:
1171:Wayback Machine
1162:
1158:
1149:
1140:
1136:
1129:
1125:
1118:
1114:
1106:
1102:
1098:
1093:
1069:Frank P. Ramsey
1029:Jaakko Hintikka
1014:Michael Dummett
999:
983:
943:dependent types
871:
857:
851:
815:
789:
774:
734:
724:
703:
697:hypothesis test
689:Model selection
669:
644:
634:
622:
600:
582:species, using
553:mortality rates
545:
513:
493:
395:
385:
379:
349:model selection
249:
233:
229:
200:
196:
182:
178:
174:
156:
151:
143:
135:
118:
114:
110:
106:
102:
98:
82:Alma mater
61:
55:
52:
46:
44:
36:
24:
17:
12:
11:
5:
2094:
2084:
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2048:
2043:
2038:
2033:
2028:
2023:
2018:
2013:
1999:
1998:
1992:Per Martin-Löf
1987:
1986:External links
1984:
1983:
1982:
1960:
1945:
1942:a free version
1934:
1927:
1920:
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1754:A. P. Dempster
1750:
1735:
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1664:
1657:Grenander, Ulf
1652:
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1478:Dempster, A.P.
1469:
1456:
1443:
1431:
1418:
1409:Sankhyā Ser. A
1393:
1380:
1367:
1354:
1335:(2): 119–170.
1319:
1299:
1279:
1268:(6): 602–619.
1247:
1232:
1207:
1194:
1186:Member profile
1175:
1156:
1134:
1123:
1112:
1099:
1097:
1094:
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1081:
1076:
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1041:
1036:
1034:Edmund Husserl
1031:
1026:
1021:
1016:
1011:
1006:
1004:Franz Brentano
1000:
998:
995:
982:
979:
869:Dependent type
853:Main article:
850:
847:
843:Edmund Husserl
835:Franz Brentano
788:
785:
781:Franz Brentano
773:
770:
723:
720:
668:
665:
658:notion of the
633:
630:
599:
596:
588:biological sex
512:
509:
492:
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381:Main article:
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169:
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92:Known for
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63:
62:
56:
42:
38:
37:
34:
26:
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23:Per Martin-Löf
22:
15:
9:
6:
4:
3:
2:
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2081:Living people
2079:
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2069:
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2049:
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2034:
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2029:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2008:
2006:
1997:
1993:
1990:
1989:
1980:
1979:
1974:
1973:
1968:
1966:
1961:
1958:
1957:
1953:
1951:
1946:
1943:
1939:
1935:
1932:
1928:
1925:
1921:
1918:
1916:
1911:
1908:
1904:
1902:
1897:
1894:
1891:
1889:
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1880:
1872:
1871:0-387-96872-5
1868:
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1808:
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1800:
1796:
1793:
1789:
1786:
1782:
1779:
1775:
1771:
1770:G. A. Barnard
1767:
1763:
1759:
1755:
1751:
1748:
1744:
1740:
1736:
1733:
1729:
1725:
1721:
1720:G. A. Barnard
1717:
1713:
1710:
1706:
1703:
1699:
1696:
1693:
1692:
1683:
1682:Statist. Sci.
1679:
1677:4 1965 78—102
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1389:Statist. Sci.
1384:
1378:4 1965 78–102
1377:
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1084:Alfred Tarski
1082:
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1024:Ulf Grenander
1022:
1020:
1019:Gottlob Frege
1017:
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1012:
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1009:Rudolf Carnap
1007:
1005:
1002:
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906:
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899:
895:
894:impredicative
891:
887:
884:where he met
883:
878:
876:
870:
866:
862:
856:
846:
844:
840:
839:Gottlob Frege
836:
832:
828:
824:
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814:
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782:
778:
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743:
739:
733:
729:
719:
717:
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702:
698:
694:
690:
682:
678:
677:mixture model
673:
664:
661:
657:
653:
649:
643:
639:
629:
627:
626:Ulf Grenander
621:
617:
613:
609:
605:
595:
593:
589:
585:
581:
577:
573:
569:
565:
560:
558:
554:
550:
549:bird watching
544:
540:
536:
532:
531:Bird watching
525:
522:
517:
508:
506:
502:
498:
488:
485:
481:
476:
474:
470:
469:string itself
466:
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458:
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195:
192:
188:
185:
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177:
173:
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159:
150:
147:
142:
141:Gödel Lecture
139:
134:
131:
130:
128:
124:
121:
117:
113:
109:
105:
101:
97:
94:
90:
87:
84:
80:
76:
72:
68:
64:
59:
53:(age 82)
43:
39:
32:
27:
20:
2066:Birdwatchers
1976:
1970:
1963:
1954:
1948:
1937:
1930:
1923:
1913:
1906:
1899:
1886:
1862:
1855:
1842:
1835:
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1817:
1813:
1806:
1802:
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1791:
1784:
1777:
1746:
1731:
1708:
1701:
1681:
1674:
1667:
1660:
1639:
1632:
1610:. Elsevier.
1607:
1588:
1580:
1563:
1559:
1534:. Retrieved
1529:
1520:
1495:
1489:
1472:
1464:
1459:
1451:
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1434:
1426:
1421:
1412:
1408:
1388:
1383:
1375:
1370:
1362:
1357:
1332:
1328:
1322:
1314:
1302:
1294:
1282:
1265:
1259:
1255:
1250:
1223:. Retrieved
1219:
1210:
1197:
1159:
1145:
1137:
1126:
1115:
1107:
1103:
984:
940:
935:
933:
928:
919:, a form of
916:
905:inconsistent
879:
872:
816:
809:Intuitionism
762:Donald Rubin
742:missing data
735:
728:Missing data
714:approach to
704:
681:Old Faithful
645:
623:
612:Haar measure
576:missing data
561:
557:bird ringing
546:
535:Bird ringing
523:
494:
483:
477:
475:being used.
468:
464:
450:
426:
407:
402:
396:
371:ringed birds
367:bird-watcher
364:
353:
345:missing data
330:His brother
329:
322:
225:
224:
190:Institutions
161:
2076:1942 births
1498:(1): 1–38.
1486:Rubin, D.B.
1482:Laird, N.M.
1089:Alan Turing
1059:Dag Prawitz
1054:Peter Pagin
925:predicative
913:type theory
855:Type theory
849:Type theory
543:Ornithology
429:compression
422:probability
274:probability
266:philosopher
100:Exact tests
74:Citizenship
66:Nationality
2005:Categories
1547:References
1536:2024-04-12
1415:: 369–375.
1079:Dana Scott
859:See also:
797:Entailment
791:See also:
726:See also:
701:Exact test
687:See also:
652:Kolmogorov
636:See also:
602:See also:
568:statistics
529:See also:
505:statistics
387:See also:
184:Philosophy
51:8 May 1942
47:1942-05-08
1762:D. R. Cox
1739:D. R. Cox
1225:4 October
1141:See e.g.
841:, and of
827:judgments
758:Nan Laird
620:semigroup
58:Stockholm
1829:Synthese
1774:G. Rasch
1743:G. Rasch
1728:G. Rasch
1167:Archived
997:See also
957:, LEGO,
949:and the
901:System F
898:Girard's
660:limiting
484:infinite
433:gambling
358:, under
298:Brentano
294:judgment
262:logician
250:Swedish:
60:, Sweden
1994:at the
1758:D. Basu
1724:D. Basu
1530:Members
1512:0501537
1504:2984875
1349:2986734
971:Epigram
961:, ALF,
907:due to
683:dataset
679:on the
564:biology
562:In the
465:process
442:Chaitin
437:betting
306:Husserl
259:Swedish
69:Swedish
1869:
1614:
1595:
1510:
1502:
1347:
981:Awards
973:, and
936:et al.
915:, his
867:, and
811:, and
760:, and
746:Anders
699:, and
618:, and
580:Dunlin
541:, and
521:Dunlin
482:is an
347:, and
339:, the
304:, and
268:, and
168:Fields
152:(2020)
144:(2006)
136:(1990)
126:Awards
77:Sweden
1500:JSTOR
1345:JSTOR
1293:, in
1150:(PDF)
1096:Notes
975:Idris
967:Twelf
955:NuPRL
825:, on
772:Logic
572:birds
308:. In
302:Frege
284:. In
282:logic
176:Logic
1867:ISBN
1741:and
1726:and
1612:ISBN
1593:ISBN
1227:2022
963:Agda
730:and
640:and
566:and
519:The
503:and
391:and
343:for
292:and
41:Born
1905:In
1562:".
1337:doi
1270:doi
959:Coq
817:In
752:by
570:of
478:An
2007::
1764:,
1760:,
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1730:.
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1579:,
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1484:;
1480:;
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320:.
300:,
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1952:.
1917:.
1903:.
1890:.
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1339::
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241:f
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