The Annals of Applied Probability

On the ergodicity of the adaptive Metropolis algorithm on unbounded domains

Eero Saksman and Matti Vihola

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This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen [Bernoulli 7 (2001) 223–242] for target distributions with a noncompact support. The conditions ensuring a strong law of large numbers require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab. 16 (2006) 1462–1505].

Article information

Ann. Appl. Probab., Volume 20, Number 6 (2010), 2178-2203.

First available in Project Euclid: 19 October 2010

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Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods
Secondary: 65C40: Computational Markov chains 60J27: Continuous-time Markov processes on discrete state spaces 93E15: Stochastic stability 93E35: Stochastic learning and adaptive control

Adaptive Markov chain Monte Carlo convergence ergodicity Metropolis algorithm stochastic approximation


Saksman, Eero; Vihola, Matti. On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Ann. Appl. Probab. 20 (2010), no. 6, 2178--2203. doi:10.1214/10-AAP682.

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