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More precisely Markov's theorem can be stated as follows: [2] [3] given two braids represented by elements , ′ in the braid groups ,, their closures are equivalent links if and only if ′ can be obtained from applying to a sequence of the following operations:
The theorem was named after Carl Friedrich Gauss and Andrey Markov, although Gauss' work significantly predates Markov's. [3] But while Gauss derived the result under the assumption of independence and normality, Markov reduced the assumptions to the form stated above. [4] A further generalization to non-spherical errors was given by Alexander ...
The phrase Gauss–Markov is used in two different ways: Gauss–Markov processes in probability theory The Gauss–Markov theorem in mathematical statistics (in this theorem, one does not assume the probability distributions are Gaussian.)
The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. [1] An example of a model for such a field is the Ising model.
Markov's theorem and 100 years of the uniqueness conjecture : a mathematical journey from irrational numbers to perfect matchings. New York: Springer. ISBN 978-3-319-00887-5. OCLC 853659945. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 188–189, 1996. Cusick, T. W. and Flahive, M. E. The Markov and Lagrange ...
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality.
In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then for some input it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable , then the set is decidable .
The "Markov" in "Markov decision process" refers to the underlying structure of state transitions that still follow the Markov property. The process is called a "decision process" because it involves making decisions that influence these state transitions, extending the concept of a Markov chain into the realm of decision-making under uncertainty.