Electronic Journal of Probability

Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?

Matti Vihola

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The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk Metropolis algorithm. The proposal distribution has the following time-dependent covariance matrix, at step $n+1$<em></em>, $S_n=\mathrm{Cov}(X_1,\ldots,X_n)+\varepsilon I$,<em></em> that is, the sample covariance matrix of the history of the chain plus a (small) constant $\varepsilon&gt;0$<em> </em> multiple of the identity matrix $I$<em> </em>. The lower bound on the eigenvalues of <em>$S_n$</em> induced by the factor $\varepsilon I$<em></em> is theoretically convenient, but practically cumbersome, as a good value for the parameter <em>$\varepsilon$</em> may not always be easy to choose. This article considers variants of the AM algorithm that do not explicitly bound the eigenvalues of <em>$S_n$</em> away from zero. The behaviour of <em>$S_n$</em> is studied in detail, indicating that the eigenvalues of $S_n$<em> </em> do not tend to collapse to zero in general. In dimension one, it is shown that $S_n$<em></em> is bounded away from zero if the logarithmic target density is uniformly continuous. For a modification of the AM algorithm including an additional fixed component in the proposal distribution, the eigenvalues of <em>$S_n$</em> are shown to stay away from zero with a practically non-restrictive condition. This result implies a strong law of large numbers for super-exponentially decaying target distributions with regular contours.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 2, 45-75.

Accepted: 2 January 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

adaptive Markov chain Monte Carlo Metropolis algorithm stability stochastic approximation

This work is licensed under aCreative Commons Attribution 3.0 License.


Vihola, Matti. Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?. Electron. J. Probab. 16 (2011), paper no. 2, 45--75. doi:10.1214/EJP.v16-840. https://projecteuclid.org/euclid.ejp/1464820171

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