{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T21:38:36Z","timestamp":1790804316678,"version":"4.1.0"},"reference-count":41,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2020,5,21]],"date-time":"2020-05-21T00:00:00Z","timestamp":1590019200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000265","name":"Medical Research Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,12,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Fully Bayesian inference in the presence of unequal probability sampling requires stronger structural assumptions on the data-generating distribution than frequentist semiparametric methods, but offers the potential for improved small-sample inference and convenient evidence synthesis. We demonstrate that the Bayesian exponentially tilted empirical likelihood can be used to combine the practical benefits of Bayesian inference with the robustness and attractive large-sample properties of frequentist approaches. Estimators defined as the solutions to unbiased estimating equations can be used to define a semiparametric model through the set of corresponding moment constraints. We prove Bernstein\u2013von Mises theorems which show that the posterior constructed from the resulting exponentially tilted empirical likelihood becomes approximately normal, centred at the chosen estimator with matching asymptotic variance; thus, the posterior has properties analogous to those of the estimator, such as double robustness, and the frequentist coverage of any credible set will be approximately equal to its credibility. The proposed method can be used to obtain modified versions of existing estimators with improved properties, such as guarantees that the estimator lies within the parameter space. Unlike existing Bayesian proposals, our method does not prescribe a particular choice of prior or require posterior variance correction, and simulations suggest that it provides superior performance in terms of frequentist criteria.<\/jats:p>","DOI":"10.1093\/biomet\/asaa028","type":"journal-article","created":{"date-parts":[[2020,3,24]],"date-time":"2020-03-24T20:10:27Z","timestamp":1585080627000},"page":"857-873","source":"Crossref","is-referenced-by-count":8,"title":["Inference under unequal probability sampling with the Bayesian exponentially tilted empirical likelihood"],"prefix":"10.1093","volume":"107","author":[{"given":"A","family":"Yiu","sequence":"first","affiliation":[{"name":"Medical Research Council Biostatistics Unit, School of Clinical Medicine, University of Cambridge, Robinson Way, Cambridge CB2 0SR, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R J B","family":"Goudie","sequence":"additional","affiliation":[{"name":"Medical Research Council Biostatistics Unit, School of Clinical Medicine, University of Cambridge, Robinson Way, Cambridge CB2 0SR, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B D M","family":"Tom","sequence":"additional","affiliation":[{"name":"Medical Research Council Biostatistics Unit, School of Clinical Medicine, University of Cambridge, Robinson Way, Cambridge CB2 0SR, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,5,21]]},"reference":[{"key":"2021032608192302900_B1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1111\/j.1467-985X.2005.00377.x","article-title":"Multiparameter evidence synthesis in epidemiology and medical decision-making: Current approaches","volume":"A 161","author":"Ades,","year":"2006","journal-title":"J. R. Statist. Soc."},{"key":"2021032608192302900_B2","author":"Antonelli,","year":"2019","journal-title":"Causal Inference in high dimensions: A marriage between Bayesian modeling and good frequentist properties"},{"key":"2021032608192302900_B3","volume-title":"lpSolve: Interface to \u2018Lp_solve\u2019 v. 5.5 to Solve Linear\/Integer Programs","author":"Berkelaar,","year":"2015"},{"key":"2021032608192302900_B4","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1093\/biomet\/asp033","article-title":"Improving efficiency and robustness of the double robust estimator for a population mean with incomplete data","volume":"96","author":"Cao,","year":"2009","journal-title":"Biometrika"},{"key":"2021032608192302900_B5","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1214\/13-STS461","article-title":"Oracle, multiple robust and multipurpose calibration in a missing response problem","volume":"29","author":"Chan,","year":"2014","journal-title":"Statist. Sci."},{"key":"2021032608192302900_B6","doi-asserted-by":"crossref","first-page":"1656","DOI":"10.1080\/01621459.2017.1358172","article-title":"Bayesian estimation and comparison of moment condition models","volume":"113","author":"Chib,","year":"2018","journal-title":"J. Am. Statist. Assoc."},{"key":"2021032608192302900_B7","doi-asserted-by":"crossref","first-page":"967","DOI":"10.1093\/biomet\/85.4.967","article-title":"Bartlett adjustment of empirical discrepancy statistics","volume":"85","author":"Corcoran,","year":"1998","journal-title":"Biometrika"},{"key":"2021032608192302900_B8","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1214\/aop\/1176996454","article-title":"I-divergence geometry of probability distributions and minimization problems","volume":"3","author":"Csisz\u00e1r,","year":"1975","journal-title":"Ann. Prob."},{"key":"2021032608192302900_B9","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1093\/biomet\/79.1.113","article-title":"Bootstrap likelihoods","volume":"79","author":"Davison,","year":"1992","journal-title":"Biometrika"},{"key":"2021032608192302900_B10","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1093\/biomet\/80.1.3","article-title":"Bayes and likelihood calculations from confidence intervals","volume":"80","author":"Efron,","year":"1993","journal-title":"Biometrika"},{"key":"2021032608192302900_B11","doi-asserted-by":"crossref","DOI":"10.1201\/b16018","volume-title":"Bayesian Data Analysis","author":"Gelman,","year":"2013"},{"key":"2021032608192302900_B12","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1214\/14-BA928","article-title":"Approximate Bayesian inference for doubly robust estimation","volume":"11","author":"Graham,","year":"2016","journal-title":"Bayesian Anal"},{"key":"2021032608192302900_B13","article-title":"Discussion of \u2018An essay on the logical foundations of survey sampling, part I\u2019","volume-title":"Foundations of Statistical Inference (Proc. Sympos., Univ. Waterloo, Ontario, 1970)","author":"H\u00e1jek,","year":"1971"},{"key":"2021032608192302900_B14","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1080\/01621459.1952.10483446","article-title":"A generalization of sampling without replacement from a finite universe","volume":"47","author":"Horvitz,","year":"1952","journal-title":"J. Am. Statist. Assoc."},{"key":"2021032608192302900_B15","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1214\/aos\/1033066214","article-title":"Exponential empirical likelihood is not Bartlett correctable","volume":"24","author":"Jing,","year":"1996","journal-title":"Ann. Statist."},{"key":"2021032608192302900_B16","first-page":"523","article-title":"Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data (with Discussion)","volume":"22","author":"Kang,","year":"2007","journal-title":"Statist. Sci."},{"key":"2021032608192302900_B17","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1093\/biomet\/90.2.319","article-title":"Bayesian empirical likelihood","volume":"90","author":"Lazar,","year":"2003","journal-title":"Biometrika"},{"key":"2021032608192302900_B18","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1093\/biomet\/86.1.107","article-title":"Nonparametric likelihood ratio confidence intervals","volume":"86","author":"Lee,","year":"1999","journal-title":"Biometrika"},{"key":"2021032608192302900_B19","doi-asserted-by":"crossref","DOI":"10.1002\/9780470580066","volume-title":"Complex Surveys: A Guide to Analysis using R","author":"Lumley,","year":"2010"},{"key":"2021032608192302900_B20","doi-asserted-by":"crossref","DOI":"10.2202\/1557-4679.1205","article-title":"Cutting feedback in Bayesian regression adjustment for the propensity score","volume":"6","author":"McCandless,","year":"2010","journal-title":"Int. J. Biostatist."},{"key":"2021032608192302900_B21","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1093\/biomet\/79.2.271","article-title":"Proper likelihoods for Bayesian analysis","volume":"79","author":"Monahan,","year":"1992","journal-title":"Biometrika"},{"key":"2021032608192302900_B22","doi-asserted-by":"crossref","DOI":"10.1201\/9781420036152","volume-title":"Empirical Likelihood","author":"Owen,","year":"2001"},{"key":"2021032608192302900_B23","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1093\/biomet\/93.4.943","article-title":"Multi-level modelling under informative sampling","volume":"93","author":"Pfeffermann,","year":"2006","journal-title":"Biometrika"},{"key":"2021032608192302900_B24","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1111\/j.1467-9868.2007.00579.x","article-title":"Empirical-likelihood-based inference in missing response problems and its application in observational studies","volume":"B 69","author":"Qin,","year":"2007","journal-title":"J. R. Statist. Soc."},{"key":"2021032608192302900_B25","volume-title":"R: A Language and Environment for Statistical Computing","year":"2020"},{"key":"2021032608192302900_B26","author":"Ray,","year":"2019","journal-title":"Semiparametric Bayesian causal inference"},{"key":"2021032608192302900_B27","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1111\/biom.12273","article-title":"Discussion of \u2018On Bayesian estimation of marginal structural models\u2019","volume":"71","author":"Robins,","year":"2015","journal-title":"Biometrics"},{"key":"2021032608192302900_B28","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1080\/01621459.1994.10476818","article-title":"Estimation of regression coefficients when some regressors are not always observed","volume":"89","author":"Robins,","year":"1994","journal-title":"J. Am. Statist. Assoc."},{"key":"2021032608192302900_B29","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1093\/biomet\/ass013","article-title":"Improved double-robust estimation in missing data and causal inference models","volume":"99","author":"Rotnitzky,","year":"2012","journal-title":"Biometrika"},{"key":"2021032608192302900_B30","first-page":"185","volume-title":"Handbook of Missing Data Methodology","author":"Rotnitzky,","year":"2014"},{"key":"2021032608192302900_B31","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1214\/aos\/1176345338","article-title":"The Bayesian bootstrap","volume":"9","author":"Rubin,","year":"1981","journal-title":"Ann. Statist."},{"key":"2021032608192302900_B32","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1093\/biomet\/asw025","article-title":"A Bayesian view of doubly robust causal inference","volume":"103","author":"Saarela,","year":"2016","journal-title":"Biometrika"},{"key":"2021032608192302900_B33","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1080\/01621459.1999.10473862","article-title":"Adjusting for nonignorable drop-out using semiparametric nonresponse models (with Discussion)","volume":"94","author":"Scharfstein,","year":"1999","journal-title":"J. Am. Statist. Assoc."},{"key":"2021032608192302900_B34","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1093\/biomet\/92.1.31","article-title":"Bayesian exponentially tilted empirical likelihood","volume":"92","author":"Schennach,","year":"2005","journal-title":"Biometrika"},{"key":"2021032608192302900_B35","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1214\/009053606000001208","article-title":"Point estimation with exponentially tilted empirical likelihood","volume":"35","author":"Schennach,","year":"2007","journal-title":"Ann. Statist."},{"key":"2021032608192302900_B36","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1214\/14-BA924","article-title":"Bayesian nonparametric weighted sampling inference","volume":"10","author":"Si,","year":"2015","journal-title":"Bayesian Anal."},{"key":"2021032608192302900_B37","doi-asserted-by":"crossref","DOI":"10.1007\/b13794","volume-title":"Introduction to Nonparametric Estimation","author":"Tsybakov,","year":"2009"},{"key":"2021032608192302900_B38","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511802256","volume-title":"Asymptotic Statistics","author":"van der Vaart,","year":"1998"},{"key":"2021032608192302900_B39","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1093\/biomet\/asx073","article-title":"Approximate Bayesian inference under informative sampling","volume":"105","author":"Wang,","year":"2017","journal-title":"Biometrika"},{"key":"2021032608192302900_B40","first-page":"91","article-title":"Bayesian inference for the finite population total from a heteroscedastic probability proportional to size sample","volume":"3","author":"Zanganeh,","year":"2015","journal-title":"J. Surv. Statist. Methodol."},{"key":"2021032608192302900_B41","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1111\/j.1541-0420.2012.01830.x","article-title":"Model feedback in Bayesian propensity score estimation","volume":"69","author":"Zigler,","year":"2013","journal-title":"Biometrics"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/107\/4\/857\/36735293\/asaa028.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/107\/4\/857\/36735293\/asaa028.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T02:12:51Z","timestamp":1695953571000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/107\/4\/857\/5841607"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,21]]},"references-count":41,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,5,21]]},"published-print":{"date-parts":[[2020,12,1]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asaa028","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2020,12]]},"published":{"date-parts":[[2020,5,21]]}}}