Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a

Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

, and mathematical statistician. He is internationally renowned for his work on the foundations of

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

, statistics, mathematical logic, and

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

. Since the late 1970s, Martin-Löf's publications have been mainly in

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

. In

philosophical logic Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...

, Martin-Löf has wrestled with the philosophy of

logical consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...

and

judgment Judgement (or US spelling judgment) is also known as ''adjudication'', which means the evaluation of evidence to make a decision. Judgement is also the ability to make considered decisions. The term has at least five distinct uses. Aristotle s ...

, partly inspired by the work of

Brentano Brentano is an Italian surname. Notable people with the surname include: * Antonie Brentano, philanthropist * August Brentano, bookseller * Bernard von Brentano, novelist * Christian Brentano, German writer * Clemens Brentano, poet and novelist, ...

Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...

, and

Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...

. In

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

, Martin-Löf has been active in developing

intuitionistic type theory Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician an ...

as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced

. Until his retirement in 2009, Per Martin-Löf held a joint chair for Mathematics and Philosophy at

Stockholm University Stockholm University ( sv, Stockholms universitet) is a public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, ...

.Member profile

Academia Europaea The Academia Europaea is a pan-European Academy of Humanities, Letters, Law, and Sciences. The Academia was founded in 1988 as a functioning Europe-wide Academy that encompasses all fields of scholarly inquiry. It acts as co-ordinator of Europea ...

, retrieved 2014-01-26. His brother Anders Martin-Löf 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 exponential families, the expectation-maximization method for

missing data In statistics, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data. Mis ...

, and

model selection Model selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered. However, the task can also involve the design of experiments such that the ...

. Per Martin-Löf received his PhD in 1970 from

, under

Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...

. Martin-Löf is an enthusiastic

bird-watcher Birdwatching, or birding, is the observing of birds, either as a recreational activity or as a form of citizen science. A birdwatcher may observe by using their naked eye, by using a visual enhancement device like binoculars or a telescope, by ...

; his first scientific publication was on the mortality rates of ringed birds..

Randomness and Kolmogorov complexity

In 1964 and 1965, Martin-Löf studied in Moscow under the supervision of Andrei N. Kolmogorov. He wrote a 1966 article ''The definition of random sequences'' that gave the first suitable definition of a random sequence. Earlier researchers such as

Richard von Mises Richard Edler von Mises (; 19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He held the position of Gordo ...

had attempted to formalize the notion of a test for randomness 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

theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...

to define formally the notion of a test for randomness. This contrasts with the idea of randomness in

; in that theory, no particular element of a sample space can be said to be random. Martin-Löf randomness has since been shown to admit many equivalent characterizations — in terms of

compression Compression may refer to: Physical science *Compression (physics), size reduction due to forces *Compression member, a structural element such as a column *Compressibility, susceptibility to compression * Gas compression *Compression ratio, of a ...

, randomness tests, and

gambling Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three ele ...

— 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

betting Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three elem ...

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 thesis that the definition of Martin-Löf randomness "correctly" captures the intuitive notion of randomness has been called the "Martin-Löf–

Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentine- American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer- ...

Thesis"; it is somewhat similar to the

Church–Turing thesis In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of co ...

. Following Martin-Löf's work,

algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as str ...

defines a random string as one that cannot be produced from any computer program that is shorter than the string ( Chaitin–Kolmogorov randomness); i.e. a string whose

Kolmogorov complexity In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produ ...

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 ''process'' that produces the string (e.g. flipping a coin to produce each bit will randomly produce a string), algorithmic randomness refers to the ''string itself''. Algorithmic information theory separates random from nonrandom strings in a way that is relatively invariant to the

model of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how ...

being used. An algorithmically random sequence is an ''infinite'' 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).

Mathematical statistics

Per Martin-Löf has done important research in

mathematical statistics Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...

, which (in the Swedish tradition) includes

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

and

statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

Bird-watching and sex determination

Per Martin-Löf began

bird watching Birdwatching, or birding, is the observing of birds, either as a recreational activity or as a form of citizen science. A birdwatcher may observe by using their naked eye, by using a visual enhancement device like binoculars or a telescope, by ...

in his youth and remains an enthusiastic bird-watcher. As a teenager, he published an article on estimating the

mortality rate Mortality rate, or death rate, is a measure of the number of deaths (in general, or due to a specific cause) in a particular population, scaled to the size of that population, per unit of time. Mortality rate is typically expressed in units of d ...

s of birds, using data from

bird ringing Birds are a group of warm-blooded vertebrates constituting the class Aves (), characterised by feathers, toothless beaked jaws, the laying of hard-shelled eggs, a high metabolic rate, a four-chambered heart, and a strong yet lightweight s ...

, in a Swedish zoological journal: This paper was soon cited in leading international journals, and this paper continues to be cited. In the

biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...

and

birds Birds are a group of warm-blooded vertebrates constituting the class Aves (), characterised by feathers, toothless beaked jaws, the laying of hard-shelled eggs, a high metabolic rate, a four-chambered heart, and a strong yet lightweigh ...

, there are several problems of

. Martin-Löf's first paper discussed the problem of estimating the mortality rates of the

Dunlin The dunlin (''Calidris alpina'') is a small wader, formerly sometimes separated with the other "stints" in the genus ''Erolia''. The English name is a dialect form of "dunling", first recorded in 1531–1532. It derives from ''dun'', "dull brow ...

species, using

capture-recapture Mark and recapture is a method commonly used in ecology to estimate an animal population's size where it is impractical to count every individual. A portion of the population is captured, marked, and released. Later, another portion will be captur ...

methods. The problem of determining the

biological sex Sex is the phenotypic trait, trait that determines whether a sexual reproduction, sexually reproducing animal or plant produces male or female gametes. Male plants and animals produce smaller mobile gametes (spermatozoa, sperm, pollen), while ...

of a bird, which is extremely difficult for humans, is one of the first examples in Martin-Löf's lectures on

statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...

Probability on algebraic structures

Martin-Löf wrote a licenciate thesis on probability on algebraic structures, particularly semigroups, while a student of

Ulf Grenander Ulf Grenander (23 July 1923 – 12 May 2016) was a Swedish statistician and professor of applied mathematics at Brown University. His early research was in probability theory, stochastic processes, time series analysis, and statistical theory (p ...

at Stockholm University.

Statistical models

Martin-Löf developed innovative approaches to

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 statistica ...

. In his paper "On Tables of Random Numbers",

Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...

observed that the

frequency probability Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of a sequence, limit of its relative Frequency_(statistics), frequency in many trials (the long-run probability). Probabi ...

notion of the

limiting In electronics, a limiter is a circuit that allows signals below a specified input power or level to pass unaffected while attenuating (lowering) the peaks of stronger signals that exceed this threshold. Limiting is a type of dynamic range comp ...

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.

Model selection and hypothesis testing

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 led to a notion of "repetitive structure" and a novel treatment of sufficient statistics, in which one-parameter

exponential families In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...

were characterized. He provided a category-theoretic approach to nested statistical models, 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).

Expectation maximization method for exponential families

Martin-Löf's student, Rolf Sundberg, developed a detailed analysis of the expectation-maximization (EM) method for estimation using data from exponential families, especially with

. Sundberg credits a formula, later known as the Sundberg formula, to previous manuscripts of the Martin-Löf brothers, Per and

Anders Anders is a male name in Scandinavian languages and Fering North Frisian, an equivalent of the Greek Andreas ("manly") and the English Andrew. It originated from Andres via metathesis. In Sweden, Anders has been one of the most common names fo ...

. Many of these results reached the international scientific community through the 1976 paper on the

expectation maximization Expectation or Expectations may refer to: Science * Expectation (epistemic) * Expected value, in mathematical probability theory * Expectation value (quantum mechanics) * Expectation–maximization algorithm, in statistics Music * ''Expectati ...

(EM) method by Arthur P. Dempster,

Nan Laird Nan McKenzie Laird (born September 18, 1943) is the Harvey V. Fineberg Professor of Public Health, Emerita in Biostatistics at the Harvard T.H. Chan School of Public Health. She served as Chair of the Department from 1990 to 1999. She was the H ...

, and

Donald Rubin Donald is a masculine given name derived from the Gaelic name ''Dòmhnall''.. This comes from the Proto-Celtic *''Dumno-ualos'' ("world-ruler" or "world-wielder"). The final -''d'' in ''Donald'' is partly derived from a misinterpretation of the ...

, which was published in a leading international journal, sponsored by the

Royal Statistical Society The Royal Statistical Society (RSS) is an established statistical society. It has three main roles: a British learned society for statistics, a professional body for statisticians and a charity which promotes statistics for the public good. ...

Logic

Philosophical logic

, Per Martin-Löf has published papers on the theory of

, on

s, etc. He has been interested in Central-European philosophical traditions, especially of the German-language writings of

Franz Brentano Franz Clemens Honoratus Hermann Josef Brentano (; ; 16 January 1838 – 17 March 1917) was an influential German philosopher, psychologist, and former Catholic priest (withdrawn in 1873 due to the definition of papal infallibility in matters ...

Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...

, and of

Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...

Type theory

Martin-Löf has worked in

for many decades. From 1968 to '69 he worked as an assistant professor at the

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

where he met

William Alvin Howard William Alvin Howard (born 1926) is a proof theorist best known for his work demonstrating formal similarity between intuitionistic logic and the simply typed lambda calculus that has come to be known as the Curry–Howard correspondence. He has a ...

with whom he discussed issues related to the

Curry–Howard correspondence In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relati ...

. Martin-Löf's first draft article on type theory dates back to 1971. This

impredicative In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more com ...

theory generalized Girard's

System F System F (also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. System F formalizes parametric polymorph ...

. However, this system turned out to be

inconsistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...

due to

Girard's paradox In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sorts). They were both proved inconsistent by Je ...

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

type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundat ...

, his ''meaning explanation'', a form of

proof-theoretic semantics Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the propo ...

, which justifies predicative type theory as presented in his 1984 Bibliopolis book, and extended in a number of increasingly philosophical texts, such as his influential ''On the Meanings of the Logical Constants and the Justifications of the Logical Laws''. The 1984 type theory was extensional while the type theory presented in the book by Nordström ''et al.'' in 1990, which was heavily influenced by his later ideas, intensional, and more amenable to being implemented on a computer. Martin-Löf's intuitionistic type theory developed the notion of

dependent type In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers lik ...

s and directly influenced the development of the

calculus of constructions In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, ...

and the logical framework LF. A number of popular computer-based proof systems are based on type theory, for example NuPRL, LEGO,

Coq Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof ...

, ALF,

Agda Agda may refer to: * Agda (programming language), the programming language and theorem prover * Agda (Golgafrinchan), the character in ''The Hitchhiker's Guide to the Galaxy'' by Douglas Adams * Liten Agda, the heroine of a Swedish legend * Agda M ...

Twelf Twelf is an implementation of the logical framework LF developed by Frank Pfenning and Carsten Schürmann at Carnegie Mellon University. It is used for logic programming and for the formalization of programming language theory. Introduction At i ...

Epigram An epigram is a brief, interesting, memorable, and sometimes surprising or satirical statement. The word is derived from the Greek "inscription" from "to write on, to inscribe", and the literary device has been employed for over two mille ...

, and Idris.

Awards

Martin-Löf is a member of the

Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special ...

and of the

Notes

References

Bird watching and missing data

* *

George A. Barnard George Alfred Barnard (23 September 1915 – 9 August 2002) was a British statistician known particularly for his work on the foundations of statistics and on quality control. Biography George Barnard was born in Walthamstow, Lon ...

"Gone Birdwatching"
''New Scientist'', 4 December 1999, magazine issue 2215. * *

Probability foundations

* Per Martin-Löf. "The Definition of Random Sequences." ''Information and Control'', 9(6): 602–619, 1966. * Li, Ming and Vitányi, Paul, ''An Introduction to Kolmogorov Complexity and Its Applications'', Springer, 1997

Probability on algebraic structures, following Ulf Grenander

* Grenander, Ulf. ''Probability on Algebraic Structures''. (Dover reprint) * Martin-Löf, P. The continuity theorem on a locally compact group. ''Teor. Verojatnost. i Primenen.'' 10 1965 367–371. * Martin-Löf, Per. Probability theory on discrete semigroups. ''Z. Wahrscheinlichkeitstheorie und Verw. Gebiete'' 4 1965 78—102 * Nitis Mukhopadhyay. "A Conversation with Ulf Grenander". ''Statist. Sci.'' Volume 21, Number 3 (2006), 404–426.

Statistics foundations

* Anders Martin-Löf. 1963. "Utvärdering av livslängder i subnanosekundsområdet" ("Evaluation of lifetimes in time-lengths below one nanosecond"). ("Sundberg formula", according to Sundberg 1971) * Per Martin-Löf. 1966. ''Statistics from the point of view of statistical mechanics''. Lecture notes, Mathematical Institute, Aarhus University. ("Sundberg formula" credited to Anders Martin-Löf, according to Sundberg 1971) * Per Martin-Löf. 1970. ''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.'' Stockholm University. * Martin-Löf, P. "Exact tests, confidence regions and estimates", with a discussion by

A. W. F. Edwards Anthony William Fairbank Edwards, FRS (born 1935) is a British statistician, geneticist and evolutionary biologist. He is the son of the surgeon Harold C. Edwards, and brother of medical geneticist John H. Edwards. He has sometimes been called ...

, G. A. Barnard, D. A. Sprott, O. Barndorff-Nielsen,

D. Basu Debabrata Basu (5 July 1924 – 24 March 2001) was an Indian statistician who made fundamental contributions to the foundations of statistics. Basu invented simple examples that displayed some difficulties of likelihood-based statistics and ...

and G. Rasch. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 121–138. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. * Martin-Löf, P. Repetitive structures and the relation between canonical and microcanonical distributions in statistics and statistical mechanics. With a discussion by D. R. Cox and G. Rasch and a reply by the author. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 271–294. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. * 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, A. P. Dempster,

, D. R. Cox,

, D. A. Sprott, G. A. Barnard, O. Barndorff-Nielsen, J. D. Kalbfleisch and G. Rasch and a reply by the author. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 1–42. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. * 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. ''Scand. J. Statist.'' 1 (1974), no. 1, 3—18. * Sverdrup, Erling. "Tests without power." ''Scand. J. Statist.'' 2 (1975), no. 3, 158–160. * Martin-Löf, Per Reply to Erling Sverdrup's polemical article: ``Tests without power'' (''Scand. J. Statist.'' 2 (1975), no. 3, 158–160). ''Scand. J. Statist.'' 2 (1975), no. 3, 161–165. * Sverdrup, Erling. A rejoinder to: ``Tests without power'' (''Scand. J. Statist.'' 2 (1975), 161—165) by P. Martin-Löf. ''Scand. J. Statist.'' 4 (1977), no. 3, 136—138. * Martin-Löf, P. Exact tests, confidence regions and estimates. Foundations of probability and statistics. II. ''Synthese'' 36 (1977), no. 2, 195—206. * Rolf Sundberg. 1971. ''Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable''. Dissertation, Institute for Mathematical Statistics, Stockholm University. * Sundberg, Rolf. Maximum likelihood theory for incomplete data from an exponential family. ''Scand. J. Statist.'' 1 (1974), no. 2, 49—58. * 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. * 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. * Höglund, Thomas. The exact estimate — a method of statistical estimation. ''Z. Wahrscheinlichkeitstheorie und Verw. Gebiete'' 29 (1974), 257—271. * Lauritzen, Steffen L. ''Extremal families and systems of sufficient statistics''. Lecture Notes in Statistics, 49. Springer-Verlag, New York, 1988. xvi+268 pp.

Foundations of mathematics, logic, and computer science

* Per Martin-Löf. A theory of types. Preprint, Stockholm University, 1971. * Per Martin-Löf.
An intuitionistic theory of types
'' 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. * 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. * Per Martin-Löf.
Constructive mathematics and computer programming
'' In ''Logic, Methodology and Philosophy of Science VI, 1979''. Eds. Cohen, et al. North-Holland, Amsterdam. pp. 153–175, 1982. * Per Martin-Löf.
Intuitionistic type theory
'' (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984. * Per Martin-Löf. ''Philosophical implications of type theory'', Unpublished notes, 1987? * Per Martin-Löf. ''Substitution calculus'', 1992. Notes from a lecture given in Göteborg. * Bengt Nordström, Kent Petersson, and Jan M. Smith. ''Programming in Martin-Löf's Type Theory''. Oxford University Press, 1990. (The book is out of print, bu
a free version
has been made available.) * Per Martin-Löf.
On the Meanings of the Logical Constants and the Justifications of the Logical Laws
'' ''

Nordic Journal of Philosophical Logic Nordic most commonly refers to: * Nordic countries, written in plural as Nordics, the northwestern European countries, including Scandinavia, Fennoscandia and the North Atlantic * Scandinavia, a cultural, historical and ethno-linguistic region ...

'', 1(1): 11–60, 1996. * Per Martin-Löf.
Logic and Ethics
'' In T. Piecha and P. Schroeder-Heister, editors,
Proof-Theoretic Semantics: Assessment and Future Perspectives. Proceedings of the Third Tübingen Conference on Proof-Theoretic Semantics, 27–30 March 2019
', pages 227-235. URI:
http://dx.doi.org/10.15496/publikation-35319
'. University of Tübingen 2019.

External links

* {{DEFAULTSORT:Martin-Lof, Per Members of the Royal Swedish Academy of Sciences Members of Academia Europaea Mathematical logicians Swedish logicians Swedish statisticians 20th-century Swedish philosophers 21st-century Swedish philosophers Swedish information theorists 20th-century Swedish mathematicians 21st-century Swedish mathematicians Stockholm University faculty Birdwatchers Swedish ornithologists 1942 births Living people Tarski lecturers Gödel Lecturers

Randomness and Kolmogorov complexity

Mathematical statistics

Bird-watching and sex determination

Probability on algebraic structures

Statistical models

Model selection and hypothesis testing

Expectation maximization method for exponential families

Logic

Philosophical logic

Type theory

Awards

See also

Notes

References

Bird watching and missing data

Probability foundations

Probability on algebraic structures, following Ulf Grenander

Statistics foundations

Foundations of mathematics, logic, and computer science

External links