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Sufficient conditions for ergodicity and convergence of MH, SA, and EM algorithms. (English) Zbl 1068.65017

The authors investigate the ergodicity of a large class of Monte Carlo Markov chain (MCMC) algorithms, defined for Markov chains taking values on the general state space \(E \subset\mathbb R^d\) and having the transition kernel of the type \[ P(x, A) = P(X_{n+1}\in A | X_n = x) = \int_A p(x, y)\nu(dy) + r(x)\mathbf 1_A(x), \] for \(A\) in a Borel \(\sigma\)-field, \(p(x, \cdot)\) a subprobability density, and \(r(x)\) the rejection probability.
The aim of this article is to discuss variations of Doeblin’s ergodicity principle when the state space \(E\) is either denumerable or arbitrary, and to establish sufficient conditions for the ergodicity of Markov chains with the considered transition kernel. Then this result is proved to be useful when applied to the convergence of three classes of MCMC-type algorithms: Metropolis-Hastings (MH), simulated annealing (SA) and expectation maximization (EM).

MSC:

65C40 Numerical analysis or methods applied to Markov chains
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65C05 Monte Carlo methods
60J22 Computational methods in Markov chains
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