Construct · 1 fields · 3 citations

Markov chain: a standalone construct

One name. One construct. A landmark on the map.

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Markovchain1 CONSTRUCTMarkov chainMARKOV · MEMORYLESS

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Probability Theory

A stochastic process whose next state depends only on the current state

Andrey Markov introduced the idea in 1906 while extending the law of large numbers to sequences of dependent events, defining what is now called a Markov chain: a sequence of random states in which the probability of the next state depends only on the current state and not on the earlier history, a property known as memorylessness or the Markov property. Andrey Kolmogorov's 1931 study of the analytic methods of probability placed such processes on rigorous foundations and carried the idea from discrete steps into continuous time. In the modern formulation, treated at length in J. R. Norris's 1997 text, a Markov chain is specified by a set of states and a matrix of transition probabilities giving the chance of moving from each state to every other in one step. What makes the construct a landmark is its reach: the same memoryless structure models physical diffusion, queues, population genetics, language, web-page ranking, and the Monte Carlo sampling used across the sciences, and it yields tractable long-run quantities such as stationary distributions and expected hitting times.[1][2][3]

Unit
A sequence of states linked by transition probabilities, carrying the memoryless (Markov) property
Recovery means
Not applicable
Measured by
Estimated from observed state transitions as a transition probability matrix (or, in continuous time, a rate matrix), then fitted and tested against sequence data

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Nearby constructs

state4 fields · 6 citationsjingle · one word, four constructsentropy6 fields · 12 citationsjingle · one name, six constructstransition3 fields · 3 citationsjingle · one word, three constructs

References

  1. Markov, A. A. (1906). Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga [Extension of the law of large numbers to quantities depending on each other]. Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 15, 135–156.
  2. Kolmogoroff, A. (1931). Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen, 104(1), 415–458. https://doi.org/10.1007/BF01457949
  3. Norris, J. R. (1997). Markov chains. Cambridge University Press. https://doi.org/10.1017/CBO9780511810633

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