Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound? |
| 2. | Creator | Author's name, affiliation, country | Matti Vihola; University of Jyväskylä; Finland |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | adaptive Markov chain Monte Carlo; Metropolis algorithm; stability; stochastic approximation |
| 3. | Subject | Subject classification | 65C40; 60J27; 93E15; 93E35 |
| 4. | Description | 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 $n+1$, $S_n=\mathrm{Cov}(X_1,\ldots,X_n)+\varepsilon I$, that is, the sample covariance matrix of the history of the chain plus a (small) constant $\varepsilon>0$ multiple of the identity matrix $I$ . The lower bound on the eigenvalues of $S_n$ induced by the factor $\varepsilon I$ is theoretically convenient, but practically cumbersome, as a good value for the parameter $\varepsilon$ may not always be easy to choose. This article considers variants of the AM algorithm that do not explicitly bound the eigenvalues of $S_n$ away from zero. The behaviour of $S_n$ is studied in detail, indicating that the eigenvalues of $S_n$ do not tend to collapse to zero in general. In dimension one, it is shown that $S_n$ 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 $S_n$ 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. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | Academy of Finland; Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation;Finnish Centre of Excellence in Analysis and Dynamics Research; Finnish Graduate School in Stochastics and Statistics |
| 7. | Date | (YYYY-MM-DD) | 2011-01-02 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/840 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v16-840 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 16 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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