Per Erik Rutger Martin-Löf (; Swedish: [ˈmǎʈːɪn ˈløːv]; born 8 May 1942) is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin-Löf's publications have been mainly in logic. In philosophical logic, Martin-Löf has wrestled with the philosophy of logical consequence and judgment, partly inspired by the work of Brentano, Frege, and Husserl. In mathematical logic, Martin-Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced computer science. Until his retirement in 2009, Per Martin-Löf held a joint chair for Mathematics and Philosophy at Stockholm University. His brother Anders Martin-Löf was 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, and model selection. Per Martin-Löf received his PhD in 1970 from Stockholm University, under Andrey Kolmogorov. Martin-Löf is an enthusiastic bird-watcher; 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 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 to define formally the notion of a test for randomness. This contrasts with the idea of randomness in probability; 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, randomness tests, and gambling — 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 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 Thesis"; it is somewhat similar to the Church–Turing thesis. Following Martin-Löf's work, algorithmic information theory 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 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 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, which (in the Swedish tradition) includes probability theory and statistics.
Bird-watching and sex determination
Per Martin-Löf began bird watching in his youth and remains an enthusiastic bird-watcher. As a teenager, he published an article on estimating the mortality rates of birds, using data from bird ringing, in a Swedish zoological journal: This paper was soon cited in leading international journals, and this paper continues to be cited. In the biology and statistics of birds, there are several problems of missing data. Martin-Löf's first paper discussed the problem of estimating the mortality rates of the Dunlin species, using capture-recapture methods. The problem of determining the sex of a bird, which, for some species, is extremely difficult for humans, is one of the first examples in Martin-Löf's lectures on statistical models.
Probability on algebraic structures
Martin-Löf wrote a licenciate thesis on probability on algebraic structures, particularly semigroups, while a student of Ulf Grenander at Stockholm University.
Statistical models
Martin-Löf developed innovative approaches to statistical theory. In his paper "On Tables of Random Numbers", Kolmogorov observed that the frequency probability notion of the limiting 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
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