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Kolmogorov's criterion defines the condition for a Markov chain or continuous-time Markov chain to be time-reversible. Time reversal of numerous classes of stochastic processes has been studied, including Lévy processes, [3] stochastic networks (Kelly's lemma), [4] birth and death processes, [5] Markov chains, [6] and piecewise deterministic ...
A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equations [13] =, where P ij is the Markov transition probability from state i to state j, i.e. P ij = P(X t = j | X t − 1 = i), and π i and π j are the equilibrium probabilities of being in states i and j, respectively ...
Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain (DTMC), [11] but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention.
Reversible Markov chains are common in Markov chain Monte Carlo (MCMC) approaches because the detailed balance equation for a desired distribution π necessarily implies that the Markov chain has been constructed so that π is a steady-state distribution.
A chain is said to be reversible if the reversed process is the same as the forward process. ... is by first finding its embedded Markov chain (EMC). Strictly ...
In computational statistics, reversible-jump Markov chain Monte Carlo is an extension to standard Markov chain Monte Carlo (MCMC) methodology, introduced by Peter Green, which allows simulation (the creation of samples) of the posterior distribution on spaces of varying dimensions. [1]
Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,
An alternative proof is possible by considering the reversed process and noting that the M/M/1 queue is a reversible stochastic process. [7] Consider the figure. By Kolmogorov's criterion for reversibility, any birth-death process is a reversible Markov chain. Note that the arrival instants in the forward Markov chain are the departure instants ...