{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T02:02:54Z","timestamp":1780452174802,"version":"3.54.1"},"reference-count":64,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/100000006","name":"United States Office of Naval Research","doi-asserted-by":"publisher","award":["N000142112510"],"award-info":[{"award-number":["N000142112510"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01ES028804"],"award-info":[{"award-number":["R01ES028804"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01ES035625"],"award-info":[{"award-number":["R01ES035625"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,2,13]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Gibbs posteriors are proportional to a prior distribution multiplied by an exponentiated loss function, with a key tuning parameter that weights the information in the loss relative to the prior and provides control of posterior uncertainty. Gibbs posteriors provide a principled framework for likelihood-free Bayesian inference; however, in many situations, the inclusion of a single tuning parameter inevitably leads to poor uncertainty quantification. In particular, regardless of the value of the parameter, credible regions are far from attaining nominal frequentist coverage, even in large samples. We propose a sequential extension to Gibbs posteriors to address this problem. We prove that the proposed sequential posterior exhibits concentration and satisfies a Bernstein\u2013von Mises theorem, which holds under easily verifiable conditions in Euclidean space and on manifolds. As a by-product, we obtain the first Bernstein\u2013von Mises theorem for traditional likelihood-based Bayesian posteriors on manifolds. All methods are illustrated with an application to principal component analysis.<\/jats:p>","DOI":"10.1093\/biomet\/asag020","type":"journal-article","created":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T12:37:27Z","timestamp":1774096647000},"source":"Crossref","is-referenced-by-count":0,"title":["Sequential Gibbs posteriors with applications to principal component analysis"],"prefix":"10.1093","volume":"113","author":[{"given":"Steven","family":"Winter","sequence":"first","affiliation":[{"name":"Department of Statistical Science, Duke University , Box 90251 , Durham, North Carolina 27707, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Omar","family":"Melikechi","sequence":"additional","affiliation":[{"name":"Department of Statistical Science, Duke University , Box 90251 , Durham, North Carolina 27707, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David B","family":"Dunson","sequence":"additional","affiliation":[{"name":"Department of Statistical Science, Duke University , Box 90251 , Durham, North Carolina 27707, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"2026060221062790100_asag020-B1","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/978-3-642-40020-9_35","volume-title":"Geometric Science of Information (Lecture Notes Comp. Sci","author":"Betancourt","year":"2013"},{"key":"2026060221062790100_asag020-B2","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.jspi.2021.10.003","article-title":"Gibbs posterior inference on multivariate quantiles","volume":"218","author":"Bhattacharya","year":"2022","journal-title":"J. Statist. Plan. Infer"},{"key":"2026060221062790100_asag020-B3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jspi.2013.08.001","article-title":"Statistics on manifolds and landmarks based image analysis: a nonparametric theory with applications","volume":"145","author":"Bhattacharya","year":"2014","journal-title":"J. Statist. Plan. Infer"},{"key":"2026060221062790100_asag020-B4","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1111\/rssb.12158","article-title":"A general framework for updating belief distributions","volume":"78","author":"Bissiri","year":"2016","journal-title":"J. R. Statist. Soc. B"},{"key":"2026060221062790100_asag020-B5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v076.i01","article-title":"Stan: a probabilistic programming language","volume":"76","author":"Carpenter","year":"2017","journal-title":"J. Statist. Software"},{"key":"2026060221062790100_asag020-B6","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1214\/18-AOS1692","article-title":"Statistics on the Stiefel manifold: theory and applications","volume":"47","author":"Chakraborty","year":"2019","journal-title":"Ann. Statist."},{"key":"2026060221062790100_asag020-B7","doi-asserted-by":"crossref","first-page":"104998","DOI":"10.1016\/j.jmva.2022.104998","article-title":"Consistency and asymptotic normality of M-estimates of scatter on Grassmann manifolds","volume":"190","author":"Ciobotaru","year":"2022","journal-title":"J. Mult. Anal"},{"key":"2026060221062790100_asag020-B8","first-page":"1","volume-title":"Proc. 33rd Conf. Uncertainty Artif. Intel","author":"Dziugaite","year":"2017"},{"key":"2026060221062790100_asag020-B9","doi-asserted-by":"crossref","first-page":"3360","DOI":"10.1214\/18-AOS1781","article-title":"A smeary central limit theorem for manifolds with application to high-dimensional spheres","volume":"47","author":"Eltzner","year":"2019","journal-title":"Ann. Statist."},{"key":"2026060221062790100_asag020-B10","first-page":"2247","volume-title":"2017 IEEE Int. Conf. Acoust. Speech Sig. Proces. (ICASSP)","author":"Elvira","year":"2017"},{"key":"2026060221062790100_asag020-B11","first-page":"737","volume-title":"Proc. 26th Int. Conf. Neural Info. Proces. Syst., vol","author":"Fox","year":"2012"},{"key":"2026060221062790100_asag020-B12","first-page":"353","volume-title":"Proc. 26th Ann. Int. Conf. Mach. Learn","author":"Germain","year":"2009"},{"key":"2026060221062790100_asag020-B13","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. R. Statist. Soc. B"},{"key":"2026060221062790100_asag020-B14","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1214\/17-BA1085","article-title":"Inconsistency of Bayesian inference for misspecified linear models, and a proposal for repairing it","volume":"12","author":"Gr\u00fcnwald","year":"2017","journal-title":"Bayesian Anal."},{"key":"2026060221062790100_asag020-B15","first-page":"391","volume-title":"SMF 2018: Congr\u00e8s de la Soci\u00e9t\u00e9 Math\u00e9matique de France","author":"Guedj","year":"2019"},{"key":"2026060221062790100_asag020-B16","doi-asserted-by":"crossref","first-page":"1342","DOI":"10.1214\/aop\/1176990748","article-title":"Asymptotic properties of the bootstrap for heavy-tailed distributions","volume":"18","author":"Hall","year":"1990","journal-title":"Ann. Prob"},{"key":"2026060221062790100_asag020-B17","volume-title":"The Bootstrap and Edgeworth Expansion","author":"Hall","year":"2013"},{"key":"2026060221062790100_asag020-B18","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1214\/15-BA989","article-title":"The general projected normal distribution of arbitrary dimension: modeling and Bayesian inference","volume":"12","author":"Hernandez-Stumpfhauser","year":"2017","journal-title":"Bayesian Anal."},{"key":"2026060221062790100_asag020-B19","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1198\/jcgs.2009.07177","article-title":"Simulation of the matrix Bingham\u2013von Mises\u2013Fisher distribution, with applications to multivariate and relational data","volume":"18","author":"Hoff","year":"2009","journal-title":"J. Comp. Graph. Statist"},{"key":"2026060221062790100_asag020-B20","author":"Holbrook","year":"2016"},{"key":"2026060221062790100_asag020-B21","first-page":"497","article-title":"Assigning a value to a power likelihood in a general Bayesian model","volume":"104","author":"Holmes","year":"2017","journal-title":"Biometrika"},{"key":"2026060221062790100_asag020-B22","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1002\/wcs.1203","article-title":"Multidimensional scaling","volume":"4","author":"Hout","year":"2013","journal-title":"WIREs Cogn. Sci"},{"key":"2026060221062790100_asag020-B23","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1111\/rssb.12336","article-title":"Unbiased Markov chain Monte Carlo methods with couplings","volume":"82","author":"Jacob","year":"2020","journal-title":"J. R. Statist. Soc. B"},{"key":"2026060221062790100_asag020-B24","doi-asserted-by":"crossref","first-page":"1560","DOI":"10.3150\/19-BEJ1176","article-title":"Random orthogonal matrices and the Cayley transform","volume":"26","author":"Jauch","year":"2020","journal-title":"Bernoulli"},{"key":"2026060221062790100_asag020-B25","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1080\/10618600.2020.1859382","article-title":"Monte Carlo simulation on the Stiefel manifold via polar expansion","volume":"30","author":"Jauch","year":"2021","journal-title":"J. Comp. Graph. Statist"},{"key":"2026060221062790100_asag020-B26","doi-asserted-by":"crossref","first-page":"2207","DOI":"10.1214\/07-AOS547","article-title":"Gibbs posterior for variable selection in high-dimensional classification and data mining","volume":"36","author":"Jiang","year":"2008","journal-title":"Ann. Statist."},{"key":"2026060221062790100_asag020-B27","first-page":"1","article-title":"Can we trust the bootstrap in high-dimension?","volume":"19","author":"Karoui","year":"2016","journal-title":"J. Mach. Learn. Res"},{"key":"2026060221062790100_asag020-B28","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/01621459.2015.1123632","article-title":"A multi-resolution approximation for massive spatial datasets","volume":"112","author":"Katzfuss","year":"2017","journal-title":"J. Am. Statist. Assoc"},{"key":"2026060221062790100_asag020-B29","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1214\/11-BJPS141","article-title":"Limit theorems for empirical Fr\u00e9chet means of independent and non-identically distributed manifold-valued random variables","volume":"25","author":"Kendall","year":"2011","journal-title":"Braz. J. Prob. Statist"},{"key":"2026060221062790100_asag020-B30","author":"Kent","year":"2013"},{"key":"2026060221062790100_asag020-B31","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1137\/07070111X","article-title":"Tensor decompositions and applications","volume":"51","author":"Kolda","year":"2009","journal-title":"SIAM Rev."},{"key":"2026060221062790100_asag020-B32","doi-asserted-by":"crossref","first-page":"3236","DOI":"10.1175\/2008JAMC1763.1","article-title":"A cautionary note on the use of nonparametric bootstrap for estimating uncertainties in extreme-value models","volume":"47","author":"Kysel\u1ef3","year":"2008","journal-title":"J. Appl. Meteor. Climatol"},{"key":"2026060221062790100_asag020-B33","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1038\/44565","article-title":"Learning the parts of objects by non-negative matrix factorization","volume":"401","author":"Lee","year":"1999","journal-title":"Nature"},{"key":"2026060221062790100_asag020-B34","first-page":"535","article-title":"Bayesian nonparametric inference on the Stiefel manifold","volume":"27","author":"Lin","year":"2017","journal-title":"Statist. Sinica"},{"key":"2026060221062790100_asag020-B35","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1093\/biomet\/asz006","article-title":"General Bayesian updating and the loss-likelihood bootstrap","volume":"106","author":"Lyddon","year":"2019","journal-title":"Biometrika"},{"key":"2026060221062790100_asag020-B36","first-page":"1","volume-title":"Handbook of Statistics","author":"Martin","year":"2022"},{"key":"2026060221062790100_asag020-B37","first-page":"7598","article-title":"Asymptotic normality, concentration, and coverage of generalized posteriors","volume":"22","author":"Miller","year":"2021","journal-title":"J. Mach. Learn. Res"},{"key":"2026060221062790100_asag020-B38","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s10543-023-00979-7","article-title":"A fast randomized algorithm for computing the null space","volume":"63","author":"Nakatsukasa","year":"2023","journal-title":"BIT Numer. Math"},{"key":"2026060221062790100_asag020-B39","first-page":"5949","volume-title":"Proc. 31st Int. Conf. Neural Info. Proces. Syst","author":"Neyshabur","year":"2017"},{"key":"2026060221062790100_asag020-B40","first-page":"849","volume-title":"Proc. 15th Int. Conf. Neural Info. Proces. Syst","author":"Ng","year":"2001"},{"key":"2026060221062790100_asag020-B41","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1080\/10618600.2014.914946","article-title":"A multiresolution Gaussian process model for the analysis of large spatial datasets","volume":"24","author":"Nychka","year":"2015","journal-title":"J. Comp. Graph. Statist"},{"key":"2026060221062790100_asag020-B42","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/BF02294847","article-title":"Concise formulas for the standard errors of component loading estimates","volume":"67","author":"Ogasawara","year":"2002","journal-title":"Psychometrika"},{"key":"2026060221062790100_asag020-B43","doi-asserted-by":"crossref","first-page":"2982","DOI":"10.1214\/19-AOS1918","article-title":"Inference for spherical location under high concentration","volume":"48","author":"Paindaveine","year":"2020","journal-title":"Ann. Statist."},{"key":"2026060221062790100_asag020-B44","author":"Peruzzi","year":"2018"},{"key":"2026060221062790100_asag020-B45","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s11222-014-9503-z","article-title":"Cuts in Bayesian graphical models","volume":"25","author":"Plummer","year":"2015","journal-title":"Statist. Comp"},{"key":"2026060221062790100_asag020-B46","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1016\/j.jmva.2009.06.010","article-title":"On the distribution of penalized maximum likelihood estimators: the LASSO, SCAD, and thresholding","volume":"100","author":"P\u00f6tscher","year":"2009","journal-title":"J. Mult. Anal"},{"key":"2026060221062790100_asag020-B47","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1214\/20-BA1202","article-title":"Bayesian inference over the Stiefel manifold via the Givens representation","volume":"16","author":"Pourzanjani","year":"2021","journal-title":"Bayesian Anal."},{"key":"2026060221062790100_asag020-B48","doi-asserted-by":"publisher","DOI":"10.24432\/C53W3X","volume-title":"Communities and Crime","author":"Redmond","year":"2009"},{"key":"2026060221062790100_asag020-B49","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1093\/biomet\/asad004","article-title":"A generalized Bayes framework for probabilistic clustering","volume":"110","author":"Rigon","year":"2023","journal-title":"Biometrika"},{"key":"2026060221062790100_asag020-B50","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1214\/aoms\/1177729586","article-title":"A stochastic approximation method","volume":"22","author":"Robbins","year":"1951","journal-title":"Ann. Math. Statist."},{"key":"2026060221062790100_asag020-B51","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1080\/01621459.1985.10478123","article-title":"Qualms about bootstrap confidence intervals","volume":"80","author":"Schenker","year":"1985","journal-title":"J. Am. Statist. Assoc"},{"key":"2026060221062790100_asag020-B52","first-page":"471","article-title":"Fast Bayesian shape matching using geometric algorithms","volume":"8","author":"Schmidler","year":"2007","journal-title":"Bayesian Statist"},{"key":"2026060221062790100_asag020-B53","first-page":"379","article-title":"Some limit theorems for joint distributions","volume":"23","author":"Sethuraman","year":"1961","journal-title":"Sankhya A"},{"key":"2026060221062790100_asag020-B54","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1093\/biomet\/asy054","article-title":"Calibrating general posterior credible regions","volume":"106","author":"Syring","year":"2019","journal-title":"Biometrika"},{"key":"2026060221062790100_asag020-B55","doi-asserted-by":"crossref","first-page":"1498","DOI":"10.1214\/19-AOS1856","article-title":"Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image","volume":"48","author":"Syring","year":"2020","journal-title":"Ann. Statist."},{"key":"2026060221062790100_asag020-B56","first-page":"466","volume-title":"Proc. 28th Int. Conf. Algorithmic Learn. Theory","author":"Thiemann","year":"2017"},{"key":"2026060221062790100_asag020-B57","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1080\/10618600.2021.2000420","article-title":"Learning subspaces of different dimensions","volume":"31","author":"Thomas","year":"2022","journal-title":"J. Comp. Graph. Statist"},{"key":"2026060221062790100_asag020-B58","first-page":"1166","article-title":"On asymptotically optimal confidence regions and tests for high-dimensional models","volume":"42","author":"Van de Geer","year":"2014","journal-title":"Project Euclid"},{"key":"2026060221062790100_asag020-B59","first-page":"1462","volume-title":"Proc. 30th Int. Conf. Mach. Learn","author":"Wang","year":"2013"},{"key":"2026060221062790100_asag020-B60","first-page":"391","article-title":"Estimation of principal components and related models by iterative least squares","volume":"36","author":"Wold","year":"1966","journal-title":"J. Mult. Anal"},{"key":"2026060221062790100_asag020-B61","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1214\/21-BA1302","article-title":"A comparison of learning rate selection methods in generalized Bayesian inference","volume":"18","author":"Wu","year":"2023","journal-title":"Bayesian Anal."},{"key":"2026060221062790100_asag020-B62","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1111\/rssb.12026","article-title":"Confidence intervals for low dimensional parameters in high dimensional linear models","volume":"76","author":"Zhang","year":"2014","journal-title":"J. R. Statist. Soc. B"},{"key":"2026060221062790100_asag020-B63","doi-asserted-by":"crossref","first-page":"2180","DOI":"10.1214\/009053606000000704","article-title":"From $ \\varepsilon $-entropy to KL-entropy: analysis of minimum information complexity density estimation","volume":"34","author":"Zhang","year":"2006","journal-title":"Ann. Statist."},{"key":"2026060221062790100_asag020-B64","doi-asserted-by":"crossref","first-page":"1307","DOI":"10.1109\/TIT.2005.864439","article-title":"Information-theoretic upper and lower bounds for statistical estimation","volume":"52","author":"Zhang","year":"2006","journal-title":"IEEE Trans. Info. Theory"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asag020\/67502119\/asag020.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/2\/asag020\/67502119\/asag020.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/2\/asag020\/67502119\/asag020.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T01:06:42Z","timestamp":1780448802000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/doi\/10.1093\/biomet\/asag020\/8539919"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"references-count":64,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2,13]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asag020","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026]]},"published":{"date-parts":[[2026]]},"article-number":"asag020"}}