Weak convergence and optimal scaling of random walk Metropolis algorithms
Open accessAbstract
This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to $\infty$. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain converges to the appropriate limiting Langevin diffusion process. The limiting diffusion approximation admits a straightforward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions. The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Assuming the posterior PDF is multivariate normal, the optimal acceptance rate of proposals is approximately 0.23.
“Assuming that the posterior PDF is a multivariate normal distribution the optimal acceptance rate of proposals is ≈ 0 . 23 [25].”
From Joining forces of Bayesian and frequentist methodology: a study for inference in the presence of non-identifiability (2013) · cites this paper as [25]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
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 · see citing works
Doob, J. L., & Doob, J. L. (1953) · CERN Document Server (European Organization for Nuclear Research) · Cited by 8,964 · see citing works
Markov Processes: Characterization and Convergence.
Abakuks, A., Ethier, S. N., & Kurtz, T. G. (1987) · Biometrics · doi:10.2307/2531839 · Cited by 4,471 · see citing works
Equation of state calculations by fast computing machines
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953) · doi:10.2172/4390578 · Cited by 1,701 · see citing works
Bayesian Computation and Stochastic Systems
Besag, J. E., Green, P., Higdon, D., & Mengersen, K. L. (1995) · Statistical Science · doi:10.1214/ss/1177010123 · Cited by 943 · see citing works
Efficient Metropolis Jumping Rules
Gelman, A., Roberts, G. O., & Gilks, W. R. (1996) · doi:10.1093/oso/9780198523567.003.0038 · Cited by 814 · see citing works
Self-citationRates 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 570 · see citing works
Spatial Statistics and Bayesian Computation
Besag, J. E., & Green, P. J. (1993) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1993.tb01467.x · Cited by 541 · see citing works
A dynamical phase transition in a caricature of a spin glass
Frigessi, A., & den Hollander, F. (1994) · Journal of Statistical Physics · doi:10.1007/bf02186873 · Cited by 5 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
MCMC Using Hamiltonian Dynamics
Neal, R. M. (2011) · Handbook of Markov Chain Monte Carlo · doi:10.1201/b10905-6 · Cited by 2,574 · see citing works
Raue, A., Kreutz, C., Theis, F. J., & Timmer, J. (2013) · Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences · doi:10.1098/rsta.2011.0544 · Cited by 119 · see citing works
Using <i>Gaia</i> DR2 to constrain local dark matter density and thin dark disk
Buch, J., Leung, S. C. (., & Fan, J. (2019) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2019/04/026 · Cited by 110 · see citing works
- Authors
- Roberts, G. O., Gelman, A., & Gilks, W. R.
- Affiliations
- University of Cambridge (GB)
- Venue
- The Annals of Applied Probability
- Publisher
- Institute of Mathematical Statistics
- Published
- 1997-02-01
- Volume
- 7
- Issue
- 1
- Pages
- 110-120
- ISSN
- 1050-5164 (ISSN-L)
- 2168-8737
- OpenAlex
- W2030911724
- Cited by
- 1,693
- References
- 9
- Citations by year
- 2026: 64 · 2025: 64 · 2024: 71 · 2023: 77 · 2022: 82 · 2021: 119 · 2020: 150 · 2019: 101 · 2018: 100 · 2017: 93 · 2016: 102 · 2015: 100 · 2014: 89 · 2013: 81 · 2012: 87
- FWCI
- 7.50
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Applied Probability (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Funders
- National Science Foundation (9457824, 9404305)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Simulation Techniques and Applications
- Keywords
- optimal scaling, weak convergence, Langevin diffusion, Markov chain Monte Carlo, acceptance rate, high-dimensional asymptotics
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
