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On the containment condition for adaptive Markov chain Monte Carlo algorithms. (English) Zbl 1225.60130

The authors study in detail the containment condition for adaptive Markov chain Monte Carlo (MCMC) algorithms. The importance of this condition lies in the fact that in combination with dimishing adaptation conditions it implies ergodicity of the MCMC algorithm on unbounded state spaces, i.e., the convergence of the marginals of the iterates to the target distribution in the total-variation metric. Whereas the dimishing adaptation condition is in most cases easily ensured by an appropriate construction of the algorithm, the containment condition is usually more difficult to prove. The aim of the authors is to present sufficient conditions for the containment condition to hold. These are expected to be easier to verify for a particular MCMC algorithm than the containment condition, and thus, the application of MCMC methods is facilitated.
In a first analysis, the authors discuss the necessity of the containment and the dimishing adaptation condition. Neither of them is strictly necessary, however, they show that, under an additional assumption, containment is necessary. Next, it is proven that, under additional assumptions, ergodicity is implied by a summable adaptive condition. Further, it is shown that containment is implied by a simultaneously geometrically ergodic condition as well as a weaker simultaneously polynomially ergodic condition. In the final part of the paper, the authors discuss conditions such that containment holds for the adaptive Metropolis-Hastings algorithm.

MSC:

60J22 Computational methods in Markov chains
60J05 Discrete-time Markov processes on general state spaces
65C40 Numerical analysis or methods applied to Markov chains
65C05 Monte Carlo methods

Software:

AMCMC; Grapham
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