The DOI matches its Crossref record; 9 of the 9 references it cites were found in a registry or an index.
Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms
Abstract
We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and Metropolis algorithms. For those based on random walk candidate distributions, we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution π. By phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly-used statistical forms. From these results we develop central limit theorems for the Metropolis algorithm. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded away from unity, are also given; these show that the negative-definiteness property is not redundant.
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Cites
Equation of State Calculations by Fast Computing Machines
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953) · The Journal of Chemical Physics · doi:10.1063/1.1699114 · Cited by 37,192
Monte Carlo sampling methods using Markov chains and their applications.
Hastings, W. K. (1970) · Biometrika · doi:10.1093/biomet/57.1.97 · Cited by 15,195
Markov Chains and Stochastic Stability
Meyn, S. P., & Tweedie, R. L. (1993) · Springer eBooks · doi:10.1007/978-1-4471-3267-7 · Cited by 5,041
Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods
Smith, A. F. M., & Roberts, G. O. (1993) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1993.tb01466.x · Cited by 1,713
Self-citationBayesian Statistics without Tears: A Sampling–Resampling Perspective
Smith, A. F. M., & Gelfand, A. E. (1992) · The American Statistician · doi:10.1080/00031305.1992.10475856 · Cited by 1,003
Rates of convergence of the Hastings and Metropolis algorithms
Mengersen, K. L., & Tweedie, R. L. (1996) · The Annals of Statistics · doi:10.1214/aos/1033066201 · Cited by 553
Spatial statistics and Bayesian computation (with discussion)
Besag, J. E., & Green, P. J. (1993) · Bristol Research (University of Bristol) · Cited by 119
Asymptotic Behavior of the Gibbs Sampler
Chan, K. (1993) · Journal of the American Statistical Association · doi:10.1080/01621459.1993.10594324 · Cited by 76
Exploring Posterior Distributions Using Markov Chains
Tierney, L. (1992) · Cited by 65
Cited by
Rue, H., Martino, S., & Chopin, N. (2009) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.1467-9868.2008.00700.x · Cited by 5,279
An introduction to MCMC for machine learning.
Andrieu, C., de Freitas, N., Doucet, A., & Jordan, M. I. (2003) · Machine Learning · doi:10.1023/a:1020281327116 · Cited by 2,451
Markov chain Monte Carlo method and its application
Brooks, S. (1998) · Journal of the Royal Statistical Society: Series D (The Statistician) · doi:10.1111/1467-9884.00117 · Cited by 843
The EM Algorithm—an Old Folk-song Sung to a Fast New Tune
Meng, X.-L., & Van Dyk, D. (1997) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/1467-9868.00082 · Cited by 802
General state space Markov chains and MCMC algorithms
Roberts, G. O., & Rosenthal, J. S. (2004) · Probability Surveys · doi:10.1214/154957804100000024 · Cited by 734
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Metadata
- Authors
- Roberts, G.
- Venue
- Biometrika
- Publisher
- Oxford University Press (OUP)
- Published
- 1996-03-01
- Volume
- 83
- Issue
- 1
- Pages
- 95-110
- ISSN
- 0006-3444 (ISSN-L)
- 1464-3510
- OpenAlex
- W2082856185
- Cited by
- 417
- References
- 9
- FWCI
- 11.08
- Type
- journal article
- Language
- en
- Open access
- closed
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Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
