The DOI matches its Crossref record; 69 of the 69 references it cites were found in a registry or an index.
General state space Markov chains and MCMC algorithms
Open accessAbstract
This paper surveys various results about Markov chains on general (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for Metropolis-Hastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.
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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
Stochastic relaxation, gibbs distributions, and the bayesian restoration of images.
Geman, S., & Geman, D. (1984) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.1984.4767596 · Cited by 17,949
Inference from iterative simulation using multiple sequences
Gelman, A., & Rubin, D. B. (1992) · Statistical Science · doi:10.1214/ss/1177011136 · Cited by 16,627
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
Doob, J. L., & Doob, J. L. (1953) · CERN Document Server (European Organization for Nuclear Research) · Cited by 9,053
Markov Chain Monte Carlo in Practice
(1995) · Chapman & Hall/CRC eBooks · doi:10.1201/b14835 · Cited by 7,757
Sampling-based approaches to calculating marginal densities
Gelfand, A. E., & Smith, A. F. M. (1990) · Journal of the American Statistical Association · doi:10.1080/01621459.1990.10476213 · Cited by 6,606
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
Markov Processes: Characterization and Convergence.
Abakuks, A., Ethier, S. N., & Kurtz, T. G. (1987) · Biometrics · doi:10.2307/2531839 · Cited by 4,233
Markov Processes: Characterization and Convergence
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Cited by
The geometric foundations of Hamiltonian Monte Carlo
Betancourt, M., Byrne, S., Livingstone, S., & Girolami, M. (2017) · Bernoulli · doi:10.3150/16-bej810 · Cited by 452
Fixed-Width Output Analysis for Markov Chain Monte Carlo
Jones, G. L., Haran, M., Caffo, B. S., & Neath, R. (2006) · Journal of the American Statistical Association · doi:10.1198/016214506000000492 · Cited by 301
Hwang, C.-R., Hwang-Ma, S.-Y., & Sheu, S.-J. (2005) · The Annals of Applied Probability · doi:10.1214/105051605000000025 · Cited by 113
Metadata
- Authors
- Roberts, G. O., & Rosenthal, J. S.
- Venue
- Probability Surveys
- Publisher
- Institute of Mathematical Statistics
- Published
- 2004-01-01
- Volume
- 1
- Issue
- none
- Pages
- 20-71
- ISSN
- 1549-5787 (ISSN-L)
- OpenAlex
- W2142623799
- Cited by
- 734
- References
- 69
- FWCI
- 10.83
- Type
- journal article
- Language
- en
- Open access
- gold
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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.
