{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T03:06:26Z","timestamp":1782183986235,"version":"3.54.5"},"reference-count":22,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:00:00Z","timestamp":1775692800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01ES035735"],"award-info":[{"award-number":["R01ES035735"]}],"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":["R01ES032418"],"award-info":[{"award-number":["R01ES032418"]}],"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":["R01ES032418"],"award-info":[{"award-number":["R01ES032418"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2434664"],"award-info":[{"award-number":["DMS-2434664"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,4,9]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>Humans are exposed to complex mixtures of environmental pollutants rather than single chemicals, necessitating methods to quantify the health effects of such mixtures. Research on environmental mixtures provides insights into realistic exposure scenarios, informing regulatory policies that better protect public health. However, statistical challenges, including complex correlations among pollutants and nonlinear multivariate exposure-response relationships, complicate such analyses. A popular Bayesian semi-parametric Gaussian process regression framework addresses these challenges by modeling exposure-response functions with Gaussian processes and performing feature selection to manage high-dimensional exposures while accounting for confounders. Originally designed for small to moderate-sized cohort studies, this framework does not scale well to massive datasets. To address this, we propose a divide-and-conquer strategy, partitioning data, computing posterior distributions in parallel, and combining results using the generalized median. While we focus on Gaussian process models for environmental mixtures, the proposed distributed computing strategy is broadly applicable to other Bayesian models with computationally prohibitive full-sample Markov Chain Monte Carlo fitting. We apply this method to estimate associations between a mixture of ambient air pollutants and 650 000 birthweights recorded in Massachusetts during 2001\u20132012. Our results reveal negative associations between birthweight and traffic pollution markers, including elemental and organic carbon and PM$_{2.5}$, and positive associations with ozone and vegetation greenness.<\/jats:p>","DOI":"10.1093\/biomtc\/ujag071","type":"journal-article","created":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T11:29:35Z","timestamp":1775388575000},"source":"Crossref","is-referenced-by-count":1,"title":["Scalable Gaussian process regression via median posterior inference for estimating the health effects of an environmental mixture"],"prefix":"10.1093","volume":"82","author":[{"given":"Aaron","family":"Sonabend-W","sequence":"first","affiliation":[{"name":"Google Research , 1600 Amphitheatre Parkway, Mountain View, CA 94043 ,","place":["United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiangshan","family":"Zhang","sequence":"additional","affiliation":[{"name":"University of California Department of Statistics, , Davis, One Shields Avenue, Davis, CA 95616 ,","place":["United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4771-8813","authenticated-orcid":false,"given":"Edgar","family":"Castro","sequence":"additional","affiliation":[{"name":"Harvard University Department of Environmental Health, , 677 Huntington Ave, Boston, MA 02115 ,","place":["United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joel","family":"Schwartz","sequence":"additional","affiliation":[{"name":"Harvard University Department of Environmental Health, , 677 Huntington Ave, Boston, MA 02115 ,","place":["United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1808-4156","authenticated-orcid":false,"given":"Brent A","family":"Coull","sequence":"additional","affiliation":[{"name":"Harvard University Department of Biostatistics, , 677 Huntington Ave, Boston, MA 02115 ,","place":["United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3743-6855","authenticated-orcid":false,"given":"Junwei","family":"Lu","sequence":"additional","affiliation":[{"name":"Harvard University Department of Biostatistics, , 677 Huntington Ave, Boston, MA 02115 ,","place":["United States"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,5,12]]},"reference":[{"key":"2026051804113642700_bib1","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.annepidem.2011.11.004","article-title":"Estimating the health effects of exposure to multi-pollutant mixture","volume":"22","author":"Billionnet","year":"2012","journal-title":"Annals of Epidemiology"},{"key":"2026051804113642700_bib2","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1093\/biostatistics\/kxu058","article-title":"Bayesian kernel machine regression for estimating the health effects of multi-pollutant mixtures","volume":"16","author":"Bobb","year":"2015","journal-title":"Biostatistics"},{"key":"2026051804113642700_bib3","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/S0925-7721(02)00102-5","article-title":"Fast approximations for sums of distances, clustering and the Fermat\u2013Weber problem","volume":"24","author":"Bose","year":"2003","journal-title":"Comput. Geom."},{"key":"2026051804113642700_bib4","first-page":"18","article-title":"Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm","volume":"19","author":"Cardot","year":"2013","journal-title":"Bernoulli (Andover.)"},{"key":"2026051804113642700_bib5","first-page":"5","article-title":"Part 1. Statistical learning methods for the effects of multiple air pollution constituents","volume-title":"Research report - Health Effects Institute","author":"Coull","year":"2015"},{"key":"2026051804113642700_bib6","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.atmosenv.2016.02.002","article-title":"A hybrid prediction model for PM$_{2.5}$ mass and components using a chemical transport model and land use regression","volume":"131","author":"Di","year":"2016","journal-title":"Atmospheric Environment"},{"key":"2026051804113642700_bib7","doi-asserted-by":"crossref","first-page":"e047","DOI":"10.1097\/EE9.0000000000000047","article-title":"Relative toxicities of major particulate matter constituents on birthweight in Massachusetts","volume":"3","author":"Fong","year":"2019","journal-title":"Environmental Epidemiology"},{"key":"2026051804113642700_bib8","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1097\/EDE.0000000000001039","article-title":"Fine particulate air pollution and birthweight: differences in associations along the birthweight distribution","volume":"30","author":"Fong","year":"2019","journal-title":"Epidemiology (Cambridge, Mass.)"},{"key":"2026051804113642700_bib9","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1093\/aje\/kwz151","article-title":"Supplemental folate and the relationship between traffic-related air pollution and livebirth among women undergoing assisted reproduction","volume":"188","author":"Gaskins","year":"2019","journal-title":"American Journal of Epidemiology"},{"key":"2026051804113642700_bib10","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.3390\/ijerph19031378","article-title":"Powering research through innovative methods for mixtures in epidemiology (PRIME) program: novel and expanded statistical methods","volume":"19","author":"Joubert","year":"2022","journal-title":"International Journal of Environmental Research and Public Health"},{"key":"2026051804113642700_bib11","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1111\/j.1541-0420.2007.00799.x","article-title":"Semiparametric regression of multidimensional genetic pathway data: Least-squares kernel machines and linear mixed models","volume":"63","author":"Liu","year":"2007","journal-title":"Biometrics"},{"key":"2026051804113642700_bib12","doi-asserted-by":"crossref","first-page":"2308","DOI":"10.3150\/14-BEJ645","article-title":"Geometric median and robust estimation in Banach spaces","volume":"21","author":"Minsker","year":"2015","journal-title":"Bernoulli: Official Journal of the Bernoulli Society for Mathematical Statistics and Probability"},{"key":"2026051804113642700_bib13","first-page":"1","article-title":"Robust and Scalable Bayes via a Median of Subset Posterior Measures","volume":"18","author":"Minsker","year":"2017","journal-title":"Journal Of Machine Learning Research"},{"key":"2026051804113642700_bib14","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.neunet.2014.09.003","article-title":"Deep learning in neural networks: An overview","volume":"61","author":"Schmidhuber","year":"2015","journal-title":"Neural Networks"},{"key":"2026051804113642700_bib15","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1214\/15-AOS1321","article-title":"Consistency of random forests","volume":"43","author":"Scornet","year":"2015","journal-title":"The Annals of Statistics"},{"key":"2026051804113642700_bib16","first-page":"912","article-title":"WASP: Scalable Bayes via barycenters of subset posteriors","volume":"38","author":"Srivastava","year":"2015","journal-title":"Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics"},{"key":"2026051804113642700_bib17","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.envres.2012.05.007","article-title":"Ambient air pollution, birth weight and preterm birth: A systematic review and meta-analysis","volume":"117","author":"Stieb","year":"2012","journal-title":"Environmental Research"},{"key":"2026051804113642700_bib18","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression shrinkage and selection via the lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"Journal of the Royal Statistical Society. Series B, Methodological"},{"key":"2026051804113642700_bib19","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2545-2","volume-title":"Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics","author":"Vaart","year":"1996"},{"key":"2026051804113642700_bib20","first-page":"2655","article-title":"Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth","volume":"37","author":"van\u00a0der\u00a0Vaart","year":"2009","journal-title":"The Annals of Statistics"},{"key":"2026051804113642700_bib21","volume-title":"Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning series","author":"Williams","year":"2019"},{"key":"2026051804113642700_bib22","doi-asserted-by":"crossref","first-page":"119356","DOI":"10.1016\/j.envpol.2022.119356","article-title":"A review of practical statistical methods used in epidemiological studies to estimate the health effects of multi-pollutant mixture","volume":"306","author":"Yu","year":"2022","journal-title":"Environmental Pollution"}],"container-title":["Biometrics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biometrics\/article-pdf\/82\/2\/ujag071\/68273186\/ujag071.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biometrics\/article-pdf\/82\/2\/ujag071\/68273186\/ujag071.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T08:12:05Z","timestamp":1779091925000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biometrics\/article\/doi\/10.1093\/biomtc\/ujag071\/8676488"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,9]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4,9]]}},"URL":"https:\/\/doi.org\/10.1093\/biomtc\/ujag071","relation":{},"ISSN":["0006-341X","1541-0420"],"issn-type":[{"value":"0006-341X","type":"print"},{"value":"1541-0420","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,6]]},"published":{"date-parts":[[2026,4,9]]},"article-number":"ujag071"}}