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One method of finding the stationary probability distribution, π, of an ergodic continuous-time Markov chain, Q, is by first finding its embedded Markov chain (EMC). Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a jump process .
A detailed definition is given here. Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state. In the theory of manifolds, an n-manifold is irreducible if any embedded (n − 1)-sphere bounds an embedded n-ball.
An aperiodic, reversible, and irreducible Markov Chain can then be obtained using Metropolis–Hastings algorithm. Persi Diaconis and Bernd Sturmfels showed that (1) a Markov basis can be defined algebraically as an Ising model [ 2 ] and (2) any generating set for the ideal I := ker ( ψ ∗ ϕ ) {\displaystyle I:=\ker({\psi }*{\phi ...
This Markov chain is irreducible, because the ghosts can fly from every state to every state in a finite amount of time. Due to the secret passageway, the Markov chain is also aperiodic, because the ghosts can move from any state to any state both in an even and in an uneven number of state transitions.
The theorem has a natural interpretation in the theory of finite Markov chains (where it is the matrix-theoretic equivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in terms of the transition matrix of the chain; see, for example, the article on the subshift of finite type).
In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state distribution.. More precisely, a fundamental result about Markov chains is that a finite state irreducible aperiodic chain has a unique stationary distribution π and, regardless of the initial state, the time-t distribution of the chain converges to π as t tends to infinity.
In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution.
Intuitively, a stochastic matrix represents a Markov chain; the application of the stochastic matrix to a probability distribution redistributes the probability mass of the original distribution while preserving its total mass. If this process is applied repeatedly, the distribution converges to a stationary distribution for the Markov chain.