{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T09:16:50Z","timestamp":1772011010851,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T00:00:00Z","timestamp":1741219200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T00:00:00Z","timestamp":1741219200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"National Science Foundation","award":["DMS CAREER-2045068"],"award-info":[{"award-number":["DMS CAREER-2045068"]}]},{"name":"National Science Foundation","award":["CIF-1908905"],"award-info":[{"award-number":["CIF-1908905"]}]},{"DOI":"10.13039\/501100001747","name":"Hong Kong Baptist University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001747","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>The topic of robustness is experiencing a resurgence of interest in the statistical and machine learning communities. In particular, robust algorithms making use of the so-called median of means estimator were shown to satisfy strong performance guarantees for many problems, including estimation of the mean, covariance structure as well as linear regression. In this work, we propose an extension of the median of means principle to the Bayesian framework, leading to the notion of the robust posterior distribution. In particular, we (a) quantify robustness of this posterior to outliers, (b) show that it satisfies a version of the Bernstein-von Mises theorem that connects Bayesian credible sets to the traditional confidence intervals, and (c) demonstrate that our approach performs well in applications.<\/jats:p>","DOI":"10.1007\/s10994-025-06754-9","type":"journal-article","created":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T18:10:09Z","timestamp":1741284609000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Generalized median of means principle for Bayesian inference"],"prefix":"10.1007","volume":"114","author":[{"given":"Stanislav","family":"Minsker","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6370-9648","authenticated-orcid":false,"given":"Shunan","family":"Yao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,6]]},"reference":[{"issue":"1","key":"6754_CR1","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1006\/jcss.1997.1545","volume":"58","author":"N Alon","year":"1999","unstructured":"Alon, N., Matias, Y., & Szegedy, M. (1999). The space complexity of approximating the frequency moments. Journal of Computer and system sciences, 58(1), 137\u2013147.","journal-title":"Journal of Computer and system sciences"},{"issue":"5","key":"6754_CR2","doi-asserted-by":"crossref","first-page":"2766","DOI":"10.1214\/11-AOS918","volume":"39","author":"JY Audibert","year":"2011","unstructured":"Audibert, J. Y., Catoni, O., et al. (2011). Robust linear least squares regression. The Annals of Statistics, 39(5), 2766\u20132794.","journal-title":"The Annals of Statistics"},{"issue":"6","key":"6754_CR3","doi-asserted-by":"crossref","first-page":"3699","DOI":"10.1214\/20-AOS1948","volume":"48","author":"Y Baraud","year":"2020","unstructured":"Baraud, Y., Birg\u00e9, L., et al. (2020). Robust Bayes-like estimation: Rho-Bayes estimation. Annals of Statistics, 48(6), 3699\u20133720.","journal-title":"Annals of Statistics"},{"key":"6754_CR4","doi-asserted-by":"crossref","unstructured":"Bayarri, M. J., & Berger, J. O. (1994). Robust Bayesian bounds for outlier detection. Recent Advances in Statistics and Probability, 175\u2013190.","DOI":"10.1515\/9783112313961-016"},{"issue":"1","key":"6754_CR5","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1214\/18-AOS1712","volume":"47","author":"A Bhattacharya","year":"2019","unstructured":"Bhattacharya, A., Pati, D., & Yang, Y. (2019). Bayesian fractional posteriors. The Annals of Statistics, 47(1), 39\u201366.","journal-title":"The Annals of Statistics"},{"issue":"5","key":"6754_CR6","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1111\/rssb.12158","volume":"78","author":"PG Bissiri","year":"2016","unstructured":"Bissiri, P. G., Holmes, C. C., & Walker, S. G. (2016). A general framework for updating belief distributions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(5), 1103\u20131130.","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"6","key":"6754_CR7","doi-asserted-by":"crossref","first-page":"2507","DOI":"10.1214\/15-AOS1350","volume":"43","author":"C Brownlees","year":"2015","unstructured":"Brownlees, C., Joly, E., & Lugosi, G. (2015). Empirical risk minimization for heavy-tailed losses. Annals of Statistics, 43(6), 2507\u20132536.","journal-title":"Annals of Statistics"},{"issue":"4","key":"6754_CR8","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1016\/j.dss.2009.05.016","volume":"47","author":"P Cortez","year":"2009","unstructured":"Cortez, P., Cerdeira, A., Almeida, F., et al. (2009). Modeling wine preferences by data mining from physicochemical properties. Decision Support Systems, 47(4), 547\u2013553.","journal-title":"Decision Support Systems"},{"key":"6754_CR9","unstructured":"Diakonikolas, I., & Kane, D. M. (2019). Recent advances in algorithmic high-dimensional robust statistics. arXiv preprint arXiv:1911.05911."},{"key":"6754_CR10","doi-asserted-by":"crossref","unstructured":"Doksum, K. A., & Lo, A. Y. (1990). Consistent and robust Bayes procedures for location based on partial information. The Annals of Statistics, 443\u2013453.","DOI":"10.1214\/aos\/1176347510"},{"key":"6754_CR11","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.3758\/s13423-017-1417-2","volume":"26","author":"G Dutilh","year":"2019","unstructured":"Dutilh, G., Annis, J., Brown, S. D., et al. (2019). The quality of response time data inference: A blinded, collaborative assessment of the validity of cognitive models. Psychonomic Bulletin Review, 26, 1051\u20131069.","journal-title":"Psychonomic Bulletin Review"},{"issue":"2","key":"6754_CR12","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s10463-014-0499-0","volume":"68","author":"A Ghosh","year":"2016","unstructured":"Ghosh, A., & Basu, A. (2016). Robust Bayes estimation using the density power divergence. Annals of the Institute of Statistical Mathematics, 68(2), 413\u2013437.","journal-title":"Annals of the Institute of Statistical Mathematics"},{"key":"6754_CR13","unstructured":"Hoff, P. D. (2007). Extending the rank likelihood for semiparametric copula estimation. The Annals of Applied Statistics, 265\u2013283."},{"issue":"1","key":"6754_CR14","first-page":"1593","volume":"15","author":"MD Hoffman","year":"2014","unstructured":"Hoffman, M. D., & Gelman, A. (2014). The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian monte carlo. Journal of Machine Learning Research, 15(1), 1593\u20131623.","journal-title":"Journal of Machine Learning Research"},{"issue":"3","key":"6754_CR15","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1007\/s11749-014-0360-z","volume":"23","author":"G Hooker","year":"2014","unstructured":"Hooker, G., & Vidyashankar, A. N. (2014). Bayesian model robustness via disparities. Test, 23(3), 556\u2013584.","journal-title":"Test"},{"key":"6754_CR16","volume-title":"Robust statistics","author":"PJ Huber","year":"2004","unstructured":"Huber, P. J. (2004). Robust statistics (Vol. 523). Wiley."},{"key":"6754_CR17","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/0304-3975(86)90174-X","volume":"43","author":"MR Jerrum","year":"1986","unstructured":"Jerrum, M. R., Valiant, L. G., & Vazirani, V. V. (1986). Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science, 43, 169\u2013188.","journal-title":"Theoretical Computer Science"},{"issue":"6","key":"6754_CR18","doi-asserted-by":"crossref","first-page":"442","DOI":"10.3390\/e20060442","volume":"20","author":"J Jewson","year":"2018","unstructured":"Jewson, J., Smith, J. Q., & Holmes, C. (2018). Principles of Bayesian inference using general divergence criteria. Entropy, 20(6), 442.","journal-title":"Entropy"},{"key":"6754_CR19","doi-asserted-by":"crossref","unstructured":"Kleijn, B., & van\u00a0der Vaart, A. (2006). Misspecification in infinite-dimensional bayesian statistics. The Annals of Statistics, 837\u2013877.","DOI":"10.1214\/009053606000000029"},{"key":"6754_CR20","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1214\/12-EJS675","volume":"6","author":"B Kleijn","year":"2012","unstructured":"Kleijn, B., & van der Vaart, A. (2012). The Bernstein-Von-Mises theorem under misspecification. Electronic Journal of Statistics, 6, 354\u2013381.","journal-title":"Electronic Journal of Statistics"},{"key":"6754_CR21","unstructured":"Knoblauch, J., Jewson, J., & Damoulas, T. (2019). Generalized variational inference: Three arguments for deriving new posteriors. arXiv preprint arXiv:1904.02063."},{"key":"6754_CR22","doi-asserted-by":"crossref","unstructured":"Laurent, B., & Massart, P. (2000). Adaptive estimation of a quadratic functional by model selection. Annals of Statistics, 1302\u20131338.","DOI":"10.1214\/aos\/1015957395"},{"issue":"2","key":"6754_CR23","doi-asserted-by":"crossref","first-page":"906","DOI":"10.1214\/19-AOS1828","volume":"48","author":"G Lecu\u00e9","year":"2020","unstructured":"Lecu\u00e9, G., & Lerasle, M. (2020). Robust machine learning by median-of-means: theory and practice. Annals of Statistics, 48(2), 906\u2013931.","journal-title":"Annals of Statistics"},{"key":"6754_CR24","unstructured":"Lerasle, M., & Oliveira, R. I. (2011). Robust empirical mean estimators. arXiv preprint arXiv:1112.3914."},{"issue":"5","key":"6754_CR25","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1007\/s10208-019-09427-x","volume":"19","author":"G Lugosi","year":"2019","unstructured":"Lugosi, G., & Mendelson, S. (2019). Mean estimation and regression under heavy-tailed distributions: A survey. Foundations of Computational Mathematics, 19(5), 1145\u20131190.","journal-title":"Foundations of Computational Mathematics"},{"issue":"3","key":"6754_CR26","doi-asserted-by":"crossref","first-page":"925","DOI":"10.4171\/jems\/937","volume":"22","author":"G Lugosi","year":"2019","unstructured":"Lugosi, G., & Mendelson, S. (2019). Risk minimization by median-of-means tournaments. Journal of the European Mathematical Society, 22(3), 925\u2013965.","journal-title":"Journal of the European Mathematical Society"},{"key":"6754_CR27","doi-asserted-by":"crossref","unstructured":"Mathieu, T., & Minsker, S. (2021). Excess risk bounds in robust empirical risk minimization. Information and Inference: A Journal of the IMA.","DOI":"10.1093\/imaiai\/iaab004"},{"key":"6754_CR28","doi-asserted-by":"crossref","unstructured":"Matsubara, T., Knoblauch, J., Briol, F. X., et\u00a0al. (2021). Robust generalised Bayesian inference for intractable likelihoods. arXiv preprint arXiv:2104.07359.","DOI":"10.1111\/rssb.12500"},{"issue":"168","key":"6754_CR29","first-page":"1","volume":"22","author":"JW Miller","year":"2021","unstructured":"Miller, J. W. (2021). Asymptotic normality, concentration, and coverage of generalized posteriors. Journal of Machine Learning Research, 22(168), 1\u201353.","journal-title":"Journal of Machine Learning Research"},{"key":"6754_CR30","unstructured":"Miller, J. W., & Dunson, D. B. (2015). Robust Bayesian inference via coarsening. arXiv preprint arXiv:1506.06101."},{"key":"6754_CR31","unstructured":"Minsker, S. (2018). Uniform bounds for robust mean estimators. arXiv preprint arXiv:1812.03523."},{"key":"6754_CR32","unstructured":"Minsker, S. (2020). Asymptotic normality of robust risk minimizers. arXiv preprint arXiv:2004.02328."},{"issue":"2","key":"6754_CR33","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1080\/03610926.2018.1543765","volume":"49","author":"T Nakagawa","year":"2020","unstructured":"Nakagawa, T., & Hashimoto, S. (2020). Robust Bayesian inference via $$\\gamma$$-divergence. Communications in Statistics-Theory and Methods, 49(2), 343\u2013360.","journal-title":"Communications in Statistics-Theory and Methods"},{"key":"6754_CR34","volume-title":"Problem complexity and method efficiency in optimization","author":"AS Nemirovsky","year":"1983","unstructured":"Nemirovsky, A. S., & Yudin, D. B. (1983). Problem complexity and method efficiency in optimization. Wiley-Interscience."},{"key":"6754_CR35","unstructured":"Nicenboim, B., Schad, D., & Vasishth, S. (2021). An introduction to bayesian data analysis for cognitive science. Under contract with Chapman and Hall\/CRC statistics in the social and behavioral sciences series."},{"issue":"4","key":"6754_CR36","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1137\/130938633","volume":"57","author":"H Owhadi","year":"2015","unstructured":"Owhadi, H., Scovel, C., & Sullivan, T. (2015). On the brittleness of bayesian inference. siam REVIEW, 57(4), 566\u2013582.","journal-title":"siam REVIEW"},{"key":"6754_CR37","unstructured":"Pacchiardi, L., & Dutta, R. (2021). Generalized bayesian likelihood-free inference using scoring rules estimators. arXiv preprint arXiv:2104.03889."},{"issue":"3","key":"6754_CR38","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1214\/14-BA926","volume":"10","author":"M Panov","year":"2015","unstructured":"Panov, M., & Spokoiny, V. (2015). Finite sample bernstein-von mises theorem for semiparametric problems. Bayesian Analysis, 10(3), 665\u2013710.","journal-title":"Bayesian Analysis"},{"key":"6754_CR39","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.neucom.2004.11.018","volume":"64","author":"M Svensen","year":"2005","unstructured":"Svensen, M., & Bishop, C. M. (2005). Robust Bayesian mixture modelling. Neurocomputing, 64, 235\u2013252.","journal-title":"Neurocomputing"},{"key":"6754_CR40","doi-asserted-by":"crossref","DOI":"10.1007\/b13794","volume-title":"Introduction to Nonparametric Estimation","author":"AB Tsybakov","year":"2009","unstructured":"Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer."},{"key":"6754_CR41","volume-title":"Asymptotic statistics,","author":"AW Van der Vaart","year":"2000","unstructured":"Van der Vaart, A. W. (2000). Asymptotic statistics, (Vol. 3). Cambridge University Press."},{"key":"6754_CR42","doi-asserted-by":"crossref","unstructured":"Van Der\u00a0Vaart, A. W., & Wellner, J. (1996). Weak convergence and empirical processes: with applications to statistics. Springer Science & Business Media.","DOI":"10.1007\/978-1-4757-2545-2"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06754-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-025-06754-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06754-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,30]],"date-time":"2025-03-30T15:14:55Z","timestamp":1743347695000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-025-06754-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,6]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["6754"],"URL":"https:\/\/doi.org\/10.1007\/s10994-025-06754-9","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,6]]},"assertion":[{"value":"16 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 March 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"No ethics approval is applicable to this article.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"No consent to participate is applicable to this article.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"No consent for publication is applicable to this article.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"115"}}