{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T12:50:18Z","timestamp":1785761418759,"version":"3.56.0"},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"3","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\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["N00014-24-1-2544"],"award-info":[{"award-number":["N00014-24-1-2544"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,7,6]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Tests of goodness of fit are used in nearly every domain where statistics is applied. One powerful and flexible approach is to sample artificial datasets that are exchangeable with the real data under the null hypothesis (but not under the alternative), as this allows the analyst to conduct a valid test using any test statistic they desire. Such sampling is typically done by conditioning on either an exact or an approximate sufficient statistic, but existing methods for doing so have significant limitations, which either preclude their use or substantially reduce their power or computational tractability for many important models. In this article, we propose to condition on samples from a Bayesian posterior distribution, which constitute a very different type of approximate sufficient statistic than those considered in previous work. Our approach, termed approximately co-sufficient sampling via Bayes, considerably expands the scope of thisflexible type of goodness-of-fit testing. We prove the approximate validity of the resulting test and demonstrate its utility on three common null models to which no existing methods apply, as well as its superior performance on models to which existing methods do apply.<\/jats:p>","DOI":"10.1093\/biomet\/asag019","type":"journal-article","created":{"date-parts":[[2026,3,29]],"date-time":"2026-03-29T10:25:56Z","timestamp":1774779956000},"source":"Crossref","is-referenced-by-count":0,"title":["Conditioning on posterior samples for flexible frequentist goodness-of-fit testing"],"prefix":"10.1093","volume":"113","author":[{"given":"Ritwik","family":"Bhaduri","sequence":"first","affiliation":[{"name":"Department of Statistics, Harvard University , One Oxford Street , Cambridge, Massachusetts 02138, U.S.A. ritwikbhaduri@g.harvard.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aabesh","family":"Bhattacharyya","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4854-8644","authenticated-orcid":false,"given":"Rina Foygel","family":"Barber","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Chicago , 5747 South Ellis Avenue , Chicago, Illinois 60637, U.S.A. aabesh@uchicago.edu \u00a0 rina@uchicago.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lucas","family":"Janson","sequence":"additional","affiliation":[{"name":"Department of Statistics, Harvard University , One Oxford Street , Cambridge, Massachusetts 02138, U.S.A. ljanson@fas.harvard.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,3,28]]},"reference":[{"issue":"1","key":"2026080308113894900_asag019-B1","doi-asserted-by":"publisher","DOI":"10.3847\/2515-5172\/ad18b5","article-title":"Spectral fit residuals as an indicator to increase model complexity","volume":"8","author":"Acharya","year":"2024","journal-title":"Res. Not. AAS"},{"key":"2026080308113894900_asag019-B2","author":"Adusumilli","year":"2018"},{"key":"2026080308113894900_asag019-B3","first-page":"131","article-title":"A survey of exact inference for contingency tables","volume":"7","author":"Agresti","year":"1992","journal-title":"Statist. Sci."},{"key":"2026080308113894900_asag019-B4","author":"Awan","year":"2020"},{"key":"2026080308113894900_asag019-B5","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511804779","volume-title":"Bayesian Reasoning and Machine Learning","author":"Barber","year":"2012"},{"key":"2026080308113894900_asag019-B6","doi-asserted-by":"crossref","first-page":"2514","DOI":"10.1214\/22-AOS2187","article-title":"Testing goodness-of-fit and conditional independence with approximate co-sufficient sampling","volume":"50","author":"Barber","year":"2022","journal-title":"Ann. Statist."},{"key":"2026080308113894900_asag019-B7","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1098\/rspa.1937.0109","article-title":"Properties of sufficiency and statistical tests","volume":"160","author":"Bartlett","year":"1937","journal-title":"Proc. R. Soc. Lond. A"},{"key":"2026080308113894900_asag019-B8","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1111\/rssb.12340","article-title":"The conditional permutation test for independence while controlling for confounders","volume":"82","author":"Berrett","year":"2020","journal-title":"J. R. Statist. Soc. B"},{"key":"2026080308113894900_asag019-B9","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1111\/rssb.12265","article-title":"Panning for gold: \u2018Model-X\u2019 knockoffs for high dimensional controlled variable selection","volume":"80","author":"Cand\u00e8s","year":"2018","journal-title":"J. R. Statist. Soc. B"},{"key":"2026080308113894900_asag019-B10","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1080\/00031305.1992.10475878","article-title":"Explaining the Gibbs sampler","volume":"46","author":"Casella","year":"1992","journal-title":"Am. Statistician"},{"key":"2026080308113894900_asag019-B11","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1080\/00031305.1995.10476177","article-title":"Understanding the Metropolis\u2013Hastings algorithm","volume":"49","author":"Chib","year":"1995","journal-title":"Am. Statistician"},{"key":"2026080308113894900_asag019-B12","first-page":"2606","author":"Chwialkowski","year":"2016"},{"key":"2026080308113894900_asag019-B13","author":"Cowell","year":"2009"},{"key":"2026080308113894900_asag019-B14","doi-asserted-by":"crossref","DOI":"10.1201\/9780203753064","volume-title":"Goodness-of-Fit Techniques","author":"D\u2019Agostino","year":"2017"},{"key":"2026080308113894900_asag019-B15","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1046\/j.1420-9101.2001.00306.x","article-title":"Selection of evolutionary models for phylogenetic hypothesis testing using parametric methods","volume":"14","author":"Emerson","year":"2001","journal-title":"J. Evol. Biol"},{"key":"2026080308113894900_asag019-B16","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1093\/biomet\/84.1.235","article-title":"Stochastic simulations conditioned on sufficient statistics","volume":"84","author":"Engen","year":"1997","journal-title":"Biometrika"},{"key":"2026080308113894900_asag019-B17","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.chaos.2014.05.005","article-title":"Goodness of fit assessment for a fractal model of stock markets","volume":"66","author":"Frezza","year":"2014","journal-title":"Chaos, Solitons & Fractals"},{"key":"2026080308113894900_asag019-B18","doi-asserted-by":"crossref","DOI":"10.1002\/9781119387596","volume-title":"Markov Chains: From Theory to Implementation and Experimentation","author":"Gagniuc","year":"2017"},{"key":"2026080308113894900_asag019-B19","author":"Gangrade","year":"2023"},{"key":"2026080308113894900_asag019-B20","doi-asserted-by":"crossref","DOI":"10.1201\/9780429258411","volume-title":"Bayesian Data Analysis","author":"Gelman","year":"1995"},{"key":"2026080308113894900_asag019-B21","doi-asserted-by":"crossref","first-page":"361","DOI":"10.2307\/2532296","article-title":"Performing the exact test of Hardy\u2013Weinberg proportion for multiple alleles","volume":"48","author":"Guo","year":"1992","journal-title":"Biometrics"},{"key":"2026080308113894900_asag019-B22","doi-asserted-by":"crossref","first-page":"1821","DOI":"10.1111\/rssb.12544","article-title":"Conditional independence testing in Hilbert spaces with applications to functional data analysis","volume":"84","author":"Lundborg","year":"2022","journal-title":"J. R. Statist. Soc. B"},{"key":"2026080308113894900_asag019-B23","doi-asserted-by":"crossref","first-page":"20881","DOI":"10.1073\/pnas.1820003116","article-title":"Sampling can be faster than optimization","volume":"116","author":"Ma","year":"2019","journal-title":"Proc. Nat. Acad. Sci"},{"key":"2026080308113894900_asag019-B24","author":"Muggeo","year":"2023"},{"key":"2026080308113894900_asag019-B25","volume-title":"R: A Language and Environment for Statistical Computing","author":"R Development Core Team","year":"2026"},{"key":"2026080308113894900_asag019-B26","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.ijar.2021.06.017","article-title":"Testing exchangeability: Fork-convexity, supermartingales and e-processes","volume":"141","author":"Ramdas","year":"2022","journal-title":"Int. J. Approx. Reason"},{"key":"2026080308113894900_asag019-B27","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1111\/rssb.12503","article-title":"Optimal thinning of MCMC output","volume":"84","author":"Riabiz","year":"2022","journal-title":"J. R. Statist. Soc. B"},{"key":"2026080308113894900_asag019-B28","author":"Saha","year":"2024"},{"key":"2026080308113894900_asag019-B29","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1093\/biomet\/asu026","article-title":"Testing independence and goodness-of-fit in linear models","volume":"101","author":"Sen","year":"2014","journal-title":"Biometrika"},{"key":"2026080308113894900_asag019-B30","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1111\/j.2517-6161.1995.tb02060.x","article-title":"Laplace approximation of high dimensional integrals","volume":"57","author":"Shun","year":"1995","journal-title":"J. R. Statist. Soc. B"},{"key":"2026080308113894900_asag019-B31","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1007\/s11009-011-9267-2","article-title":"Goodness-of-fit and sufficiency: Exact and approximate tests","volume":"14","author":"Stephens","year":"2012","journal-title":"Methodol. Comp. Appl. Prob"},{"key":"2026080308113894900_asag019-B32","first-page":"170","article-title":"Determining optimum burn-in and replacement times using Bayesian decision theory","volume":"21","author":"Stewart","year":"2009","journal-title":"IEEE Trans. Reliab"},{"key":"2026080308113894900_asag019-B33","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/978-3-642-41136-6_2","volume-title":"Empirical Inference: Festschrift in Honor of Vladimir N. Vapnik","author":"Vapnik","year":"2013"},{"key":"2026080308113894900_asag019-B34","author":"Wu","year":"2020"},{"key":"2026080308113894900_asag019-B35","author":"Xie","year":"2025"},{"key":"2026080308113894900_asag019-B36","author":"Zhu","year":"2023"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asag019\/67637139\/asag019.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/3\/asag019\/67637139\/asag019.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/113\/3\/asag019\/67637139\/asag019.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T12:11:48Z","timestamp":1785759108000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/doi\/10.1093\/biomet\/asag019\/8555607"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,7,6]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asag019","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":"asag019"}}