{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:26:21Z","timestamp":1772295981961,"version":"3.50.1"},"reference-count":69,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2014,6,3]],"date-time":"2014-06-03T00:00:00Z","timestamp":1401753600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this, geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of the appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.<\/jats:p>","DOI":"10.3390\/e16063074","type":"journal-article","created":{"date-parts":[[2014,6,3]],"date-time":"2014-06-03T11:15:50Z","timestamp":1401794150000},"page":"3074-3102","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions"],"prefix":"10.3390","volume":"16","author":[{"given":"Samuel","family":"Livingstone","sequence":"first","affiliation":[{"name":"Department of Statistical Science, University College London, Gower Street, London WC1E 6BT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mark","family":"Girolami","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Warwick, Coventry CV4 7AL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1111\/j.1467-9868.2010.00765.x","article-title":"Riemann manifold Langevin and Hamiltonian Monte Carlo methods","volume":"73","author":"Girolami","year":"2011","journal-title":"J. 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Monte Carlo Statistical Methods, Springer.","DOI":"10.1007\/978-1-4757-4145-2"},{"key":"ref_8","first-page":"1701","article-title":"Markov chains for exploring posterior distributions","volume":"22","author":"Tierney","year":"1994","journal-title":"Ann. Stat"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01210789","article-title":"Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions","volume":"104","author":"Kipnis","year":"1986","journal-title":"Commun. Math. Phys"},{"key":"ref_10","unstructured":"(2012). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_11","first-page":"7","article-title":"CODA: Convergence diagnosis and output analysis for MCMC","volume":"6","author":"Plummer","year":"2006","journal-title":"R. News"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1111\/j.1751-5823.2002.tb00178.x","article-title":"On choosing and bounding probability metrics","volume":"70","author":"Gibbs","year":"2002","journal-title":"Int. Stat. Rev"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1214\/ss\/1015346315","article-title":"Honest exploration of intractable probability distributions via Markov chain Monte Carlo","volume":"16","author":"Jones","year":"2001","journal-title":"Stat. Sci"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1214\/154957804100000051","article-title":"On the Markov chain central limit theorem","volume":"1","author":"Jones","year":"2004","journal-title":"Probab. Surv"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1214\/ss\/1177011136","article-title":"Inference from iterative simulation using multiple sequences","volume":"7","author":"Gelman","year":"1992","journal-title":"Stat. Sci"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1214\/10-STS327","article-title":"The random walk Metropolis: Linking theory and practice through a case study","volume":"25","author":"Sherlock","year":"2010","journal-title":"Stat. Sci"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"774","DOI":"10.3150\/08-BEJ176","article-title":"Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets","volume":"15","author":"Sherlock","year":"2009","journal-title":"Bernoulli"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1239\/jap\/1363784420","article-title":"Optimal scaling of the random walk Metropolis: General criteria for the 0.234 acceptance rule","volume":"50","author":"Sherlock","year":"2013","journal-title":"J. Appl. Probab"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1016\/j.spa.2012.12.001","article-title":"Advanced MCMC methods for sampling on diffusion pathspace","volume":"123","author":"Beskos","year":"2013","journal-title":"Stoch. Processes Appl"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1214\/ss\/1015346320","article-title":"Optimal scaling for various Metropolis\u2013Hastings algorithms","volume":"16","author":"Roberts","year":"2001","journal-title":"Stat. Sci"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1093\/biomet\/83.1.95","article-title":"Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms","volume":"83","author":"Roberts","year":"1996","journal-title":"Biometrika"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1214\/aos\/1033066201","article-title":"Rates of convergence of the Hastings and Metropolis algorithms","volume":"24","author":"Mengersen","year":"1996","journal-title":"Ann. Stat"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0304-4149(99)00082-4","article-title":"Geometric ergodicity of Metropolis algorithms","volume":"85","author":"Jarner","year":"2000","journal-title":"Stoch. Processes Appl"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1023\/A:1013779208892","article-title":"Geometric ergodicity of Metropolis\u2013Hastings algorithms for conditional simulation in generalized linear mixed models","volume":"3","author":"Christensen","year":"2001","journal-title":"Methodol. Comput. Appl. Probab"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s11009-007-9046-2","article-title":"Optimal scaling for random walk Metropolis on spherically constrained target densities","volume":"10","author":"Neal","year":"2008","journal-title":"Methodol. Comput. Appl. Probab"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"559","DOI":"10.3150\/bj\/1066223269","article-title":"Necessary conditions for geometric and polynomial ergodicity of random-walk-type","volume":"9","author":"Jarner","year":"2003","journal-title":"Bernoulli"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"\u00d8ksendal, B. (2003). Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-642-14394-6"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Rogers, L.C.G., and Williams, D. (2000). Diffusions, Markov Processes and Martingales: Volume 2, It\u00f4 Calculus, Cambridge University Press.","DOI":"10.1017\/CBO9781107590120"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"518","DOI":"10.2307\/1427522","article-title":"Stability of Markovian processes III: Foster\u2013Lyapunov criteria for continuous-time processes","volume":"25","author":"Meyn","year":"1993","journal-title":"Adv. Appl. Probab"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Coffey, W., Kalmykov, Y.P., and Waldron, J.T. (2004). The Langevin Equation: with Applications to Stochastic Problems in Physics, Chemistry, and Electrical Engineering, World Scientific.","DOI":"10.1142\/9789812795090"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"341","DOI":"10.2307\/3318418","article-title":"Exponential convergence of Langevin distributions and their discrete approximations","volume":"2","author":"Roberts","year":"1996","journal-title":"Bernoulli"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1023\/A:1023562417138","article-title":"Langevin diffusions and Metropolis\u2013Hastings algorithms","volume":"4","author":"Roberts","year":"2002","journal-title":"Methodol. Comput. Appl. Probab"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.spl.2014.04.002","article-title":"Langevin diffusions and the Metropolis-adjusted Langevin algorithm","volume":"91","author":"Xifara","year":"2013","journal-title":"Stat. Probab. Lett"},{"key":"ref_34","first-page":"453","article-title":"An invariant form for the prior probability in estimation problems","volume":"186","author":"Jeffreys","year":"1946","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Sci"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1214\/aos\/1176349258","article-title":"Preferred point geometry and statistical manifolds","volume":"21","author":"Critchley","year":"1993","journal-title":"Ann. Stat"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1093\/biomet\/89.1.77","article-title":"On the local geometry of mixture models","volume":"89","author":"Marriott","year":"2002","journal-title":"Biometrika"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"83","DOI":"10.2307\/1403260","article-title":"The role of differential geometry in statistical theory","volume":"54","author":"Cox","year":"1986","journal-title":"Int. Stat. Rev"},{"key":"ref_38","unstructured":"Boothby, W.M. (1986). An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Lee, J.M. (2003). Smooth Manifolds, Springer.","DOI":"10.1007\/978-0-387-21752-9_1"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Do Carmo, M.P. (1992). Riemannian Geometry, Springer.","DOI":"10.1007\/978-1-4757-2201-7"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Nash, J.F. (2002). The Essential John Nash, Princeton University Press.","DOI":"10.1515\/9781400884087"},{"key":"ref_42","unstructured":"Manton, J.H. (2013). A Primer on Stochastic Differential Geometry for Signal Processing, arXiv, 1302.0430."},{"key":"ref_43","unstructured":"Stewart, J. (2011). Multivariable Calculus, Cengage Learning."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Hsu, E.P. (2002). Stochastic Analysis on Manifolds, American Mathematical Society.","DOI":"10.1090\/gsm\/038"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"819","DOI":"10.2307\/1426661","article-title":"Time-reversible diffusions","volume":"10","author":"Kent","year":"1978","journal-title":"Adv. Appl. Probab"},{"key":"ref_46","first-page":"81","article-title":"Information and accuracy attainable in the estimation of statistical parameters","volume":"37","year":"1945","journal-title":"Bull. Calcutta Math. Soc"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1198\/106186006X100470","article-title":"Robust Markov chain Monte Carlo methods for spatial generalized linear mixed models","volume":"15","author":"Christensen","year":"2006","journal-title":"J. Comput. Graph. Stat"},{"key":"ref_48","first-page":"1308.6221","article-title":"A computational framework for infinite-dimensional Bayesian inverse problems: Part II","volume":"arXiv","author":"Petra","year":"2013","journal-title":"Stochastic Newton MCMC with application to ice sheet flow inverse problems"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Pawitan, Y. (2001). In All Likelihood: Statistical Modelling and Inference Using Likelihood, Oxford University Press.","DOI":"10.1093\/oso\/9780198507659.001.0001"},{"key":"ref_50","unstructured":"Betancourt, M. (2013). Geometric Science of Information, Springer."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1093\/imanum\/22.3.329","article-title":"Computing the nearest correlation matrix\u2014a problem from finance","volume":"22","author":"Higham","year":"2002","journal-title":"IMA J. Numer. Anal"},{"key":"ref_52","unstructured":"Sejdinovic, D., Garcia, M.L., Strathmann, H., Andrieu, C., and Gretton, A. (2013). Kernel Adaptive Metropolis\u2013Hastings, arXiv, 1307.5302."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"A1460","DOI":"10.1137\/110845598","article-title":"A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion","volume":"34","author":"Martin","year":"2012","journal-title":"SIAM J. Sci. Comput"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1098\/rsfs.2011.0051","article-title":"Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods","volume":"1","author":"Calderhead","year":"2011","journal-title":"Interface Focus"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"20110541","DOI":"10.1098\/rsta.2011.0541","article-title":"Markov chain Monte Carlo inference for Markov jump processes via the linear noise approximation","volume":"371","author":"Stathopoulos","year":"2013","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.pbiomolbio.2011.07.002","article-title":"Efficient probabilistic model personalization integrating uncertainty on data and parameters: Application to eikonal-diffusion models in cardiac electrophysiology","volume":"107","author":"Konukoglu","year":"2011","journal-title":"Prog. Biophys. Mol. Biol"},{"key":"ref_57","unstructured":"Do Carmo, M.P., and Do Carmo, M.P. (1976). Differential Geometry of Curves and Surfaces, Englewood Cliffs."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Shima, H. (2007). The Geometry of Hessian Structures, World Scientific.","DOI":"10.1142\/9789812707536"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1214\/13-STS421","article-title":"MCMC methods for functions: Modifying old algorithms to make them faster","volume":"28","author":"Cotter","year":"2013","journal-title":"Stat. Sci"},{"key":"ref_60","unstructured":"Da Prato, G., and Zabczyk, J. (2008). Stochastic Equations in Infinite Dimensions, Cambridge University Press."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.cam.2013.07.026","article-title":"Proposals which speed up function-space MCMC","volume":"262","author":"Law","year":"2014","journal-title":"J. Comput. Appl. Math"},{"key":"ref_62","unstructured":"Ottobre, M., Pillai, N.S., Pinski, F.J., and Stuart, A.M. (2013). A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit, arXiv, 1308.0543."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/0370-2693(91)90812-5","article-title":"A generalized guided Monte Carlo algorithm","volume":"268","author":"Horowitz","year":"1991","journal-title":"Phys. Lett. B"},{"key":"ref_64","unstructured":"Mardia, K.V., and Jupp, P.E. (2009). Directional Statistics, Wiley."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1111\/sjos.12036","article-title":"Geodesic Monte Carlo on embedded manifolds","volume":"40","author":"Byrne","year":"2013","journal-title":"Scand. J. Stat"},{"key":"ref_66","unstructured":"Diaconis, P., Holmes, S., and Shahshahani, M. (2013). Advances in Modern Statistical Theory and Applications: A Festschrift in Honor of Morris L. Eaton, Institute of Mathematical Statistics."},{"key":"ref_67","first-page":"188","article-title":"Discussion on \u201cRiemann manifold Langevin and Hamiltonian Monte Carlo methods\u201d (by Girolami, M. and Calderhead, B.)","volume":"73","author":"Latuszynski","year":"2011","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Capinski, M., and Kopp, P.E. (2004). Measure, Integral and Probability, Springer.","DOI":"10.1007\/978-1-4471-0645-6"},{"key":"ref_69","unstructured":"Schutz, B.F. (1984). Geometrical Methods of Mathematical Physics, Cambridge University Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/6\/3074\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:12:04Z","timestamp":1760217124000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/6\/3074"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,3]]},"references-count":69,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2014,6]]}},"alternative-id":["e16063074"],"URL":"https:\/\/doi.org\/10.3390\/e16063074","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,3]]}}}