Electronic Journal of Probability

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

Matti Vihola

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Abstract

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

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

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

Permanent link to this document
http://projecteuclid.org/euclid.ejp/1464820171

Digital Object Identifier
doi:10.1214/EJP.v16-840

Mathematical Reviews number (MathSciNet)
MR2749772

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

Keywords
adaptive Markov chain Monte Carlo Metropolis algorithm stability stochastic approximation

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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. http://projecteuclid.org/euclid.ejp/1464820171.


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