A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes.
Read more about Markov Chain: Introduction, Formal Definition, Markov Chains, Finite State Space, Reversible Markov Chain, Bernoulli Scheme, General State Space, Applications, Fitting, History
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... principle can always be solved to give the equilibrium distribution of a Markov chain (when such a distribution exists) ... For a Markov chain with state space S, transition rate from state i to j given by qij and equilibrium distribution given by, the global balance equations are given for every state i in S by Here ... For a discrete time Markov chain with transition matrix P and equilibrium distribution the global balance equation is ...
... the relationship between heat diffusion and a random walk (Markov Chain) an analogy is drawn between the diffusion operator on a manifold and a Markov transition matrix operating ... It is easy to see here that from the tuple {X,k} one can construct a reversible Markov Chain ... If has to faithfully represent a Markov matrix, then it has to be normalized by the corresponding degree matrix now represents a Markov chain ...
... The concept of the Markov chain of order L, which we essentially owe to the Russian mathematician Andrej Andreevic Markov (1907), has two drawbacks ... First, the number of parameters of the model grows exponentially with the order L of the chain ... - 1992, Willems - 1995) was the Variable Length Markov chain (Buhlmann - 1999) ...
... In Markov chain Monte Carlo, the Metropolis–Hastings algorithm (MH) can be used to sample from a probability distribution which is difficult to sample from directly ... is too small, almost all steps will be accepted, and the Markov chain will be similar to a random walk through the probability space ... In this event, the Markov Chain will not fully explore the probability space in any reasonable amount of time ...
... Andrey Markov produced the first results (1906) for these processes, purely theoretically ... Markov chains are related to Brownian motion and the ergodic hypothesis, two topics in physics which were important in the early years of the ... Seneta provides an account of Markov's motivations and the theory's early development ...
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“We are all bound to the throne of the Supreme Being by a flexible chain which restrains without enslaving us. The most wonderful aspect of the universal scheme of things is the action of free beings under divine guidance.”
—Joseph De Maistre (1753–1821)