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For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and = holds, then by summing over , the global balance equations are satisfied and is the stationary distribution of the process. [5]
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 ...
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.
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix.
In the context of a continuous-time Markov process with jumps, see Kolmogorov equations (Markov jump process). In particular, in natural sciences the forward equation is also known as master equation. In the context of a diffusion process, for the backward Kolmogorov equations see Kolmogorov backward equations (diffusion).
Markov processes can only be reversible if their stationary distributions have the property of detailed balance: (=, + =) = (=, + =). Kolmogorov's criterion defines the condition for a Markov chain or continuous-time Markov chain to be time-reversible.
A Markov arrival process is defined by two matrices, D 0 and D 1 where elements of D 0 represent hidden transitions and elements of D 1 observable transitions. The block matrix Q below is a transition rate matrix for a continuous-time Markov chain.