{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T15:47:41Z","timestamp":1791215261964,"version":"4.1.0"},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"vor","delay-in-days":7760,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Ministry of University and Technological and Scientific Research"},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["9803433"],"award-info":[{"award-number":["9803433"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2002,1,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>A Bernstein prior is a probability measure on the space of all the distribution functions on [0, 1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose densities are mixtures of known beta densities. The Bernstein prior is of interest in Bayesian nonparametric inference with continuous data. We study the consistency of the posterior from a Bernstein prior. We first show that, under mild assumptions, the posterior is weakly consistent for any distribution function P0 on [0, 1] with continuous and bounded Lebesgue density. With slightly stronger assumptions on the prior, the posterior is also Hellinger consistent. This implies that the predictive density from a Bernstein prior, which is a Bayesian density estimate, converges in the Hellinger sense to the true density (assuming that it is continuous and bounded). We also study a sieve maximum likelihood version of the density estimator and show that it is also Hellinger consistent under weak assumptions. When the order of the Bernstein polynomial, i.e. the number of components in the beta distribution mixture, is truncated, we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities. Finally, we study the behaviour of the predictive density numerically and we also study a hybrid Bayes\u2013maximum likelihood density estimator.<\/jats:p>","DOI":"10.1111\/1467-9868.00326","type":"journal-article","created":{"date-parts":[[2003,3,12]],"date-time":"2003-03-12T19:37:03Z","timestamp":1047497823000},"page":"79-100","source":"Crossref","is-referenced-by-count":96,"title":["Consistency of Bernstein Polynomial Posteriors"],"prefix":"10.1093","volume":"64","author":[{"given":"Sonia","family":"Petrone","sequence":"first","affiliation":[{"name":"Universit\u00e0 dell'Insubria , Varese , Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Larry","family":"Wasserman","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University , Pittsburgh , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2002,3,12]]},"reference":[{"key":"2023040200504750900_","author":"Altomare","year":"1994"},{"key":"2023040200504750900_","volume-title":"The exponential convergence of posterior probabilities with implications for Bayes estimators of density functions","author":"Barron","year":"1988"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s004400050210","article-title":"Risk bounds for model selection via penalization","volume":"113","author":"Barron","year":"1998","journal-title":"Prob. Theory Reltd Flds"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1214\/aos\/1018031206","article-title":"The consistency of posterior distributions in nonparametric problems","volume":"27","author":"Barron","year":"1999","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","first-page":"95","article-title":"An asymptotic property of Model Selection criteria","volume":"44","author":"Barron","year":"1995","journal-title":"IEEE Trans. Inform. Theory"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1214\/aoms\/1177699597","article-title":"Limiting behavior of posterior distributions when the model is incorrect","volume":"37","author":"Berk","year":"1966","journal-title":"Ann. Math. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1214\/aoms\/1177696967","article-title":"Consistency of a posteriori","volume":"41","year":"1970","journal-title":"Ann. Math. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1214\/aos\/1028144851","article-title":"Asymptotic behavior of Bayes estimates under possibly incorrect models","volume":"26","author":"Bunke","year":"1998","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/S0167-9473(99)00010-9","article-title":"Beta kernel estimators for density functions","volume":"31","author":"Chen","year":"1999","journal-title":"Comput. Statist. Data Anal."},{"key":"2023040200504750900_","author":"Dellacherie","year":"1982"},{"key":"2023040200504750900_","first-page":"1","article-title":"On the consistency of Bayes estimates","volume":"14","author":"Diaconis","year":"1986","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","first-page":"1317","article-title":"On the uniform consistency of Bayes estimates for multinomial probabilities","volume":"18","year":"1990","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","first-page":"2108","article-title":"Nonparametric binary regression: a Bayesian approach","volume":"21","year":"1993","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","volume-title":"Some results on the probability law of random Bernstein polynomials and applications to Bayesian Statistics (in Italian)","author":"Epifani","year":"1999"},{"key":"2023040200504750900_","first-page":"167","article-title":"Sigma algebras on spaces of probability measures","volume":"16","author":"Gaudart","year":"1989","journal-title":"Scand. J. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1214\/aos\/1015956709","article-title":"Convergence rates for the Gaussian mixture sieve","volume":"28","author":"Genovese","year":"2000","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","volume-title":"Consistency issues in Bayesian non-parametrics","author":"Ghosal","year":"1998"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0378-3758(98)00192-X","article-title":"Consistent semi-parametric Bayesian inference about a location parameter","volume":"77","year":"1999","journal-title":"J. Statist. Planng Inf."},{"key":"2023040200504750900_","first-page":"143","article-title":"Posterior consistency of Dirichlet mixtures in density estimation","volume":"27","year":"1999","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1214\/aos\/1016218228","article-title":"Convergence rates of posterior distributions","volume":"28","author":"Ghosal","year":"2000","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1080\/01621459.1995.10476592","article-title":"A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Statist. Ass."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1214\/aos\/1176346412","article-title":"On a class of Bayesian nonparametric estimates: density estimates","volume":"12","author":"Lo","year":"1984","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1214\/aos\/1176348653","article-title":"Exact mean integrated squared error","volume":"20","author":"Marron","year":"1992","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1111\/1467-9469.00155","article-title":"Random Bernstein polynomials","volume":"26","author":"Petrone","year":"1999","journal-title":"Scand. J. Statist."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"105","DOI":"10.2307\/3315494","article-title":"Bayesian density estimation using Bernstein polynomials","volume":"27","year":"1999","journal-title":"Can. J. Statist."},{"key":"2023040200504750900_","volume-title":"Nonparametric mixture priors based on an exponential random scheme","author":"Petrone","year":"2001"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/BF00535479","article-title":"On Bayes procedures","volume":"4","author":"Schwartz","year":"1965","journal-title":"Z. Wahrsch. Ver. Geb."},{"key":"2023040200504750900_","doi-asserted-by":"crossref","DOI":"10.1214\/aos\/1009210686","volume-title":"Rates of convergence of posterior distributions","author":"Shen","year":"2001"},{"key":"2023040200504750900_","author":"Silverman","year":"1986"},{"key":"2023040200504750900_","author":"Tenbusch","year":"1995"},{"key":"2023040200504750900_","author":"Van Der Vaart","year":"1998"},{"key":"2023040200504750900_","author":"Van Der Vaart","year":"1996"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","DOI":"10.1016\/B978-0-12-568002-8.50011-2","volume-title":"A Bernstein polynomial approach to density estimation","author":"Vitale","year":"1975"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1732-9_16","volume-title":"Asymptotic properties of nonparametric Bayesian procedures","author":"Wasserman","year":"1998"},{"key":"2023040200504750900_","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1214\/aos\/1176324524","article-title":"Probability inequalities for likelihood ratios and convergence rates of sieve MLE's","volume":"23","author":"Wong","year":"1995","journal-title":"Ann. Statist."},{"key":"2023040200504750900_","volume-title":"Nonparametric Bayesian density and hazard rate estimation using Bernstein polynomials","author":"Wu","year":"1999"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2F1467-9868.00326","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/64\/1\/79\/49723332\/jrsssb_64_1_79.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/64\/1\/79\/49723332\/jrsssb_64_1_79.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,23]],"date-time":"2023-04-23T10:20:30Z","timestamp":1682245230000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/64\/1\/79\/7098294"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1,1]]},"references-count":36,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2002,3,12]]},"published-print":{"date-parts":[[2002,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/1467-9868.00326","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2002,1]]},"published":{"date-parts":[[2002,1,1]]}}}