{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,11]],"date-time":"2026-10-11T14:30:23Z","timestamp":1791729023549,"version":"4.3.4"},"reference-count":65,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T00:00:00Z","timestamp":1678320000000},"content-version":"vor","delay-in-days":340,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The use of heuristics to assess the convergence and compress the output of Markov chain Monte Carlo can be sub-optimal in terms of the empirical approximations that are produced. Typically a number of the initial states are attributed to \u2018burn in\u2019 and removed, while the remainder of the chain is \u2018thinned\u2019 if compression is also required. In this paper, we consider the problem of retrospectively selecting a subset of states, of fixed cardinality, from the sample path such that the approximation provided by their empirical distribution is close to optimal. A novel method is proposed, based on greedy minimisation of a kernel Stein discrepancy, that is suitable when the gradient of the log-target can be evaluated and approximation using a small number of states is required. Theoretical results guarantee consistency of the method and its effectiveness is demonstrated in the challenging context of parameter inference for ordinary differential equations. Software is available in the Stein Thinning package in Python, R and MATLAB.<\/jats:p>","DOI":"10.1111\/rssb.12503","type":"journal-article","created":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T02:50:52Z","timestamp":1649040652000},"page":"1059-1081","source":"Crossref","is-referenced-by-count":25,"title":["Optimal Thinning of MCMC Output"],"prefix":"10.1093","volume":"84","author":[{"given":"Marina","family":"Riabiz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wilson Ye","family":"Chen","sequence":"additional","affiliation":[{"name":"University of Sydney , Sydney , Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jon","family":"Cockayne","sequence":"additional","affiliation":[{"name":"Alan Turing Institute , London , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pawel","family":"Swietach","sequence":"additional","affiliation":[{"name":"Oxford University , Oxford , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steven A.","family":"Niederer","sequence":"additional","affiliation":[{"name":"King's College London , London , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lester","family":"Mackey","sequence":"additional","affiliation":[{"name":"Microsoft Research , Cambridge , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chris. J.","family":"Oates","sequence":"additional","affiliation":[{"name":"Alan Turing Institute , London , UK , Newcastle Upon Tyne , UK"},{"name":"Newcastle University , London , UK , Newcastle Upon Tyne , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,4,3]]},"reference":[{"key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1016\/j.jcp.2015.10.045","article-title":"Anatomically accurate high resolution modeling of human whole heart electromechanics: a strongly scalable algebraic multigrid solver method for nonlinear deformation","volume":"305","author":"Augustin","year":"2016","journal-title":"Journal of Computational Physics"},{"issue":"1","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/S0047-259X(03)00079-4","article-title":"On a new multivariate two-sample test","volume":"88","author":"Baringhaus","year":"2004","journal-title":"Journal of Multivariate Analysis"},{"key":"2023030906301445500_","article-title":"A Riemann\u2013Stein kernel method","author":"Barp","year":"2021","journal-title":"Bernoulli"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9096-9","volume-title":"Reproducing Kernel Hilbert spaces in probability and statistics","author":"Berlinet","year":"2004"},{"issue":"4","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1080\/10618600.1998.10474787","article-title":"General methods for monitoring convergence of iterative simulations","volume":"7","author":"Brooks","year":"1998","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"1","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1093\/cvr\/cvv149","article-title":"Stochastic spontaneous calcium release events trigger premature ventricular complexes by overcoming electrotonic load","volume":"107","author":"Campos","year":"2015","journal-title":"Cardiovascular Research"},{"issue":"1","key":"2023030906301445500_","doi-asserted-by":"crossref","DOI":"10.18637\/jss.v076.i01","article-title":"Stan: a probabilistic programming language","volume":"76","author":"Carpenter","year":"2017","journal-title":"Journal of Statistical Software"},{"key":"2023030906301445500_","volume-title":"Stein points","author":"Chen","year":"2018"},{"key":"2023030906301445500_","volume-title":"Stein points Markov chain Monte Carlo","author":"Chen","year":"2019"},{"key":"2023030906301445500_","volume-title":"A kernel test of goodness of fit","author":"Chwialkowski","year":"2016"},{"issue":"8","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"e1007260","DOI":"10.1371\/journal.pcbi.1007260","article-title":"Arrhythmia mechanisms and spontaneous calcium release: bi-directional coupling between re-entrant and focal excitation","volume":"15","author":"Colman","year":"2019","journal-title":"PLoS Computational Biology"},{"issue":"434","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1080\/01621459.1996.10476956","article-title":"Markov chain Monte Carlo convergence diagnostics: a comparative review","volume":"91","author":"Cowles","year":"1996","journal-title":"Journal of the American Statistical Association"},{"key":"2023030906301445500_","volume-title":"On the geometry of Stein variational gradient descent","author":"Duncan","year":"2019"},{"issue":"1\u20132","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s00440-018-0860-y","article-title":"The power of online thinning in reducing discrepancy","volume":"174","author":"Dwivedi","year":"2019","journal-title":"Probability Theory and Related Fields"},{"key":"2023030906301445500_","volume-title":"Large sample analysis of the median heuristic","author":"Garreau","year":"2018"},{"issue":"410","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1080\/01621459.1990.10476213","article-title":"Sampling-based approaches to calculating marginal densities","volume":"85","author":"Gelfand","year":"1990","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2023030906301445500_","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":"Statistical Science"},{"key":"2023030906301445500_","volume-title":"Bayesian data analysis","author":"Gelman","year":"2014"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1109\/TPAMI.1984.4767596","article-title":"Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images","volume":"6","author":"Geman","year":"1984","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"4","key":"2023030906301445500_","first-page":"473","article-title":"Practical Markov chain Monte Carlo","volume":"7","author":"Geyer","year":"1992","journal-title":"Statistical Science"},{"issue":"3","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1111\/j.2517-6161.1992.tb01443.x","article-title":"Constrained Monte Carlo maximum likelihood for dependent data","volume":"54","author":"Geyer","year":"1992","journal-title":"Journal of the Royal Statistical Society, Series B"},{"issue":"2","key":"2023030906301445500_","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":"Journal of the Royal Statistical Society, Series B"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/0065-2571(65)90067-1","article-title":"Oscillatory behavior in enzymatic control process","volume":"3","author":"Goodwin","year":"1965","journal-title":"Advances in Enzyme Regulation"},{"key":"2023030906301445500_","volume-title":"Measuring sample quality with Stein's method","author":"Gorham","year":"2015"},{"key":"2023030906301445500_","volume-title":"Measuring sample quality with kernels","author":"Gorham","year":"2017"},{"key":"2023030906301445500_","volume-title":"Stochastic Stein discrepancies","author":"Gorham","year":"2020"},{"issue":"4","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1007\/s11222-015-9574-5","article-title":"Bayesian computation: a summary of the current state, and samples backwards and forwards","volume":"25","author":"Green","year":"2015","journal-title":"Statistics and Computing"},{"issue":"3","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s001800050022","article-title":"Adaptive proposal distribution for random walk Metropolis algorithm","volume":"14","author":"Haario","year":"1999","journal-title":"Computational Statistics"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","DOI":"10.1093\/biomet\/57.1.97","volume-title":"Monte Carlo sampling methods using Markov chains and their applications","author":"Hastings","year":"1970"},{"issue":"221","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1090\/S0025-5718-98-00894-1","article-title":"A generalized discrepancy and quadrature error bound","volume":"67","author":"Hickernell","year":"1998","journal-title":"Mathematics of Computation"},{"key":"2023030906301445500_","volume-title":"The reproducing Stein kernel approach for post-hoc corrected sampling","author":"Hodgkinson","year":"2020"},{"key":"2023030906301445500_","volume-title":"Random feature Stein discrepancies","author":"Huggins","year":"2018"},{"issue":"1","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1080\/00401706.2014.881749","article-title":"Sequential exploration of complex surfaces using minimum energy designs","volume":"57","author":"Joseph","year":"2015","journal-title":"Technometrics"},{"issue":"3","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1080\/00401706.2018.1552203","article-title":"Deterministic sampling of expensive posteriors using minimum energy designs","volume":"61","author":"Joseph","year":"2019","journal-title":"Technometrics"},{"key":"2023030906301445500_","volume-title":"A diffusion approach to Stein's method on Riemannian manifolds","author":"Le","year":"2020"},{"key":"2023030906301445500_","volume-title":"Stein variational gradient descent as gradient flow","author":"Liu","year":"2017"},{"key":"2023030906301445500_","volume-title":"Black-box importance sampling","author":"Liu","year":"2017"},{"key":"2023030906301445500_","volume-title":"Stein variational gradient descent: a general purpose Bayesian inference algorithm","author":"Liu","year":"2016"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","DOI":"10.1609\/aaai.v32i1.11810","volume-title":"Riemannian Stein variational gradient descent for Bayesian inference","author":"Liu","year":"2018"},{"key":"2023030906301445500_","volume-title":"A kernelized Stein discrepancy for goodness-of-fit tests and model evaluation","author":"Liu","year":"2016"},{"key":"2023030906301445500_","volume-title":"The Barker proposal: combining robustness and efficiency in gradient-based MCMC","author":"Livingstone","year":"2020"},{"issue":"82","key":"2023030906301445500_","first-page":"341","article-title":"Elements of physical biology","volume":"21","author":"Lotka","year":"1926","journal-title":"Science Progress in the Twentieth Century (1919\u20131933)"},{"issue":"4","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1023\/A:1008929526011","article-title":"WinBUGS - a Bayesian modelling framework: concepts, structure, and extensibility","volume":"10","author":"Lunn","year":"2000","journal-title":"Statistics and Computing"},{"issue":"6A","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"2562","DOI":"10.1214\/17-AOS1629","article-title":"Support points","volume":"46","author":"Mak","year":"2018","journal-title":"The Annals of Statistics"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","DOI":"10.1111\/rssb.12500","article-title":"Robust generalised Bayesian inference for intractable likelihoods","author":"Matsubara","year":"2022","journal-title":"Journal of the Royal Statistical Society Series B (Statistical Methodology)"},{"key":"2023030906301445500_","volume-title":"Markov Chains and stochastic stability","author":"Meyn","year":"2012"},{"issue":"2","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"429","DOI":"10.2307\/1428011","article-title":"Integral probability metrics and their generating classes of functions","volume":"29","author":"Muller","year":"1997","journal-title":"Advances in Applied Probability"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"14","DOI":"10.3389\/fphys.2011.00014","article-title":"Simulating human cardiac electrophysiology on clinical time-scales","volume":"2","author":"Niederer","year":"2011","journal-title":"Frontiers in Physiology"},{"issue":"2","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1038\/s41569-018-0104-y","article-title":"Computational models in cardiology","volume":"16","author":"Niederer","year":"2019","journal-title":"Nature Reviews Cardiology"},{"issue":"3","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1111\/rssb.12185","article-title":"Control functionals for Monte Carlo integration","volume":"79","author":"Oates","year":"2017","journal-title":"Journal of the Royal Statistical Society, Series B"},{"key":"2023030906301445500_","volume-title":"Super-sampling with a reservoir","author":"Paige","year":"2016"},{"key":"2023030906301445500_","volume-title":"JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling","author":"Plummer","year":"2003"},{"key":"2023030906301445500_","volume-title":"R: a language and environment for statistical computing","author":"R Core Team","year":"2020"},{"key":"2023030906301445500_","volume-title":"Monte Carlo statistical methods","author":"Robert","year":"2013"},{"issue":"4","key":"2023030906301445500_","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":"2023030906301445500_","volume-title":"Learning via Hilbert space embedding of distributions","author":"Song","year":"2008"},{"key":"2023030906301445500_","article-title":"Semi-exact control functionals from Sard's method","author":"South","year":"2021","journal-title":"Biometrika"},{"key":"2023030906301445500_","first-page":"583","volume-title":"A bound for the error in the normal approximation to the distribution of a sum of dependent random variables","author":"Stein","year":"1972"},{"key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"109645","DOI":"10.1016\/j.jbiomech.2020.109645","article-title":"Simulating ventricular systolic motion in a four-chamber heart model with spatially varying robin boundary conditions to model the effect of the pericardium","volume":"101","author":"Strocchi","year":"2020","journal-title":"Journal of Biomechanics"},{"issue":"16.10","key":"2023030906301445500_","first-page":"1249","article-title":"Testing for equal distributions in high dimension","volume":"5","author":"Sz\u00e9kely","year":"2004","journal-title":"InterStat"},{"issue":"398","key":"2023030906301445500_","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1080\/01621459.1987.10478458","article-title":"The calculation of posterior distributions by data augmentation","volume":"82","author":"Tanner","year":"1987","journal-title":"Journal of the American Statistical Association"},{"key":"2023030906301445500_","volume-title":"Optimal quantisation of probability measures using maximum mean discrepancy","author":"Teymur","year":"2021"},{"key":"2023030906301445500_","volume-title":"Revisiting the Gelman-Rubin diagnostic","author":"Vats","year":"2018"},{"key":"2023030906301445500_","first-page":"31","article-title":"Variazioni e fluttuazioni del numero d'individui in specie animali conviventi","volume":"6","author":"Volterra","year":"1926","journal-title":"Memoria della Reale Accademia Nazionale dei Lincei"},{"key":"2023030906301445500_","volume-title":"A Stein goodness-of-fit test for directional distributions","author":"Xu","year":"2020"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/rssb.12503","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/rssb.12503","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/84\/4\/1059\/49463105\/jrsssb_84_4_1059.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/84\/4\/1059\/49463105\/jrsssb_84_4_1059.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,21]],"date-time":"2024-09-21T10:41:34Z","timestamp":1726915294000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/84\/4\/1059\/7073269"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,3]]},"references-count":65,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4,3]]},"published-print":{"date-parts":[[2022,9,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12503","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2022,9]]},"published":{"date-parts":[[2022,4,3]]}}}