{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T00:51:50Z","timestamp":1775868710013,"version":"3.50.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T00:00:00Z","timestamp":1744329600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T00:00:00Z","timestamp":1744329600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62376145"],"award-info":[{"award-number":["62376145"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62406179"],"award-info":[{"award-number":["62406179"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Science and Technology Innovation Talent Team of Shanxi Province","award":["202204051002016"],"award-info":[{"award-number":["202204051002016"]}]},{"name":"Taiyuan City \"Double hundred Research action\" of the first batch project about \"Leading the Charge with Open Competition\"","award":["2024TYJB0127"],"award-info":[{"award-number":["2024TYJB0127"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s10994-025-06761-w","type":"journal-article","created":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T10:33:31Z","timestamp":1744367611000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An efficient Bayes error rate estimation method"],"prefix":"10.1007","volume":"114","author":[{"given":"Qingqiang","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fuyuan","family":"Cao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Xing","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiye","family":"Liang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,11]]},"reference":[{"issue":"2","key":"6761_CR1","first-page":"255","volume":"17","author":"J Alcal\u00e1-Fdez","year":"2011","unstructured":"Alcal\u00e1-Fdez, J., Fern\u00e1ndez, A., Luengo, J., Derrac, J., Garc\u00eda, S., S\u00e1nchez, L., & Herrera, F. (2011). Keel data-mining software tool: Data set repository, integration of algorithms and experimental analysis framework. Journal of Multiple-valued Logic and Soft Computing, 17(2), 255\u2013287.","journal-title":"Journal of Multiple-valued Logic and Soft Computing"},{"issue":"7","key":"6761_CR2","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1109\/34.777375","volume":"21","author":"A Antos","year":"1999","unstructured":"Antos, A., Devroye, L., & Gyorfi, L. (1999). Lower bounds for bayes error estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(7), 643\u2013645.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"6761_CR3","doi-asserted-by":"publisher","DOI":"10.1016\/j.is.2019.02.006","volume":"87","author":"M Aum\u00fcller","year":"2020","unstructured":"Aum\u00fcller, M., Bernhardsson, E., & Faithfull, A. (2020). Ann-benchmarks: A benchmarking tool for approximate nearest neighbor algorithms. Information Systems, 87, 101374.","journal-title":"Information Systems"},{"issue":"1","key":"6761_CR4","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1109\/34.476017","volume":"18","author":"H Avi-Itzhak","year":"1996","unstructured":"Avi-Itzhak, H., & Diep, T. (1996). Arbitrarily tight upper and lower bounds on the bayesian probability of error. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(1), 89\u201391.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"3","key":"6761_CR5","doi-asserted-by":"publisher","first-page":"580","DOI":"10.1109\/TSP.2015.2477805","volume":"64","author":"V Berisha","year":"2015","unstructured":"Berisha, V., Wisler, A., Hero, A. O., & Spanias, A. (2015). Empirically estimable classification bounds based on a nonparametric divergence measure. IEEE Transactions on Signal Processing, 64(3), 580\u2013591.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"6761_CR6","volume-title":"Pattern recognition and machine learning","author":"CM Bishop","year":"2006","unstructured":"Bishop, C. M. (2006). Pattern recognition and machine learning. Springer-Verlag."},{"issue":"1","key":"6761_CR7","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1023\/A:1010933404324","volume":"45","author":"L Breiman","year":"2001","unstructured":"Breiman, L. (2001). Random forests. Machine Learning, 45(1), 5\u201332.","journal-title":"Machine Learning"},{"issue":"6","key":"6761_CR8","doi-asserted-by":"publisher","first-page":"2337","DOI":"10.1109\/TKDE.2019.2953897","volume":"33","author":"D Cai","year":"2019","unstructured":"Cai, D. (2019). A revisit of hashing algorithms for approximate nearest neighbor search. IEEE Transactions on Knowledge and Data Engineering, 33(6), 2337\u20132348.","journal-title":"IEEE Transactions on Knowledge and Data Engineering"},{"issue":"3","key":"6761_CR9","doi-asserted-by":"publisher","first-page":"1789","DOI":"10.1214\/19-AOS1868","volume":"48","author":"TI Cannings","year":"2020","unstructured":"Cannings, T. I., Berrett, T. B., & Samworth, R. J. (2020). Local nearest neighbour classification with applications to semi-supervised learning. The Annals of Statistics, 48(3), 1789\u20131814.","journal-title":"The Annals of Statistics"},{"issue":"3","key":"6761_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1961189.1961199","volume":"2","author":"CC Chang","year":"2011","unstructured":"Chang, C. C., & Lin, C. J. (2011). Libsvm: A library for support vector machines. ACM Transactions on Intelligent Systems and Technology, 2(3), 1\u201327.","journal-title":"ACM Transactions on Intelligent Systems and Technology"},{"key":"6761_CR11","doi-asserted-by":"crossref","unstructured":"Chen, Q., Cao, F., Xing, Y., Liang, J. (2022). Instance selection: A bayesian decision theory perspective. In Proceedings of the AAAI conference on artificial intelligence, pp. 6287\u20136294.","DOI":"10.1609\/aaai.v36i6.20578"},{"issue":"8","key":"6761_CR12","doi-asserted-by":"publisher","first-page":"9639","DOI":"10.1109\/TPAMI.2023.3240194","volume":"45","author":"Q Chen","year":"2023","unstructured":"Chen, Q., Cao, F., Xing, Y., & Liang, J. (2023). Evaluating classification model against bayes error rate. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(8), 9639\u20139653.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"6761_CR13","doi-asserted-by":"crossref","unstructured":"Chen, Y., Bai, Y., Zhang, W., & Mei, T. (2019). Destruction and construction learning for fine-grained image recognition. In Proceedings of the conference on computer vision and pattern recognition (pp. 5157\u20135166).","DOI":"10.1109\/CVPR.2019.00530"},{"key":"6761_CR14","doi-asserted-by":"crossref","unstructured":"Cover, T.M. (1969). Learning in pattern recognition. Methodologies of pattern recognition, pp 111\u2013132.","DOI":"10.1016\/B978-1-4832-3093-1.50012-2"},{"issue":"3","key":"6761_CR15","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1090\/S0002-9947-1940-0002052-6","volume":"47","author":"JL Doob","year":"1940","unstructured":"Doob, J. L. (1940). Regularity properties of certain families of chance variables. Transactions of the American Mathematical Society, 47(3), 455\u2013486.","journal-title":"Transactions of the American Mathematical Society"},{"key":"6761_CR16","unstructured":"Dutta, S., Wei, D., Yueksel, H., Chen, P-Y., Liu, S., Varshney, K. (2020). Is there a trade-off between fairness and accuracy? A perspective using mismatched hypothesis testing. In International conference on machine learning, pp. 2803\u20132813."},{"key":"6761_CR17","volume-title":"Introduction to statistical pattern recognition","author":"K Fukunaga","year":"2013","unstructured":"Fukunaga, K. (2013). Introduction to statistical pattern recognition. Elsevier."},{"issue":"3","key":"6761_CR18","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1109\/TIT.1975.1055373","volume":"21","author":"K Fukunaga","year":"1975","unstructured":"Fukunaga, K., & Hostetler, L. (1975). K-nearest-neighbor bayes-risk estimation. IEEE Transactions on Information Theory, 21(3), 285\u2013293.","journal-title":"IEEE Transactions on Information Theory"},{"issue":"4","key":"6761_CR19","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1109\/TIT.1973.1055049","volume":"19","author":"K Fukunaga","year":"1973","unstructured":"Fukunaga, K., & Kessell, D. (1973). Nonparametric bayes error estimation using unclassified samples. IEEE Transactions on Information Theory, 19(4), 434\u2013440.","journal-title":"IEEE Transactions on Information Theory"},{"issue":"2","key":"6761_CR20","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1109\/34.273728","volume":"16","author":"WA Hashlamoun","year":"1994","unstructured":"Hashlamoun, W. A., Varshney, P. K., & Samarasooriya, V. (1994). A tight upper bound on the bayesian probability of error. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(2), 220\u2013224.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"3","key":"6761_CR21","doi-asserted-by":"publisher","first-page":"349","DOI":"10.4310\/SII.2009.v2.n3.a8","volume":"2","author":"T Hastie","year":"2009","unstructured":"Hastie, T., Rosset, S., Zhu, J., & Zou, H. (2009). Multi-class adaboost. Statistics and Its Interface, 2(3), 349\u2013360.","journal-title":"Statistics and Its Interface"},{"key":"6761_CR22","unstructured":"Ishida, T., Yamane, I., Charoenphakdee, N., Niu, G., & Sugiyama, M. (2023). Is the performance of my deep network too good to be true? A direct approach to estimating the bayes error in binary classification. In Proceedings of the international conference on learning representations."},{"key":"6761_CR23","doi-asserted-by":"crossref","unstructured":"Jeong, M., Cardone, M., & Dytso, A. (2024). Demystifying the optimal performance of multi-class classification. In Proceedings of the advances in neural information processing systems (pp. 31638\u201331664).","DOI":"10.52202\/075280-1374"},{"issue":"3","key":"6761_CR24","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1109\/TBDATA.2019.2921572","volume":"7","author":"J Johnson","year":"2019","unstructured":"Johnson, J., Douze, M., & J\u00e9gou, H. (2019). Billion-scale similarity search with gpus. IEEE Transactions on Big Data, 7(3), 535\u2013547.","journal-title":"IEEE Transactions on Big Data"},{"issue":"1","key":"6761_CR26","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1109\/TCOM.1967.1089532","volume":"15","author":"T Kailath","year":"1967","unstructured":"Kailath, T. (1967). The divergence and 4 bhattacharyya distance measures in signal selection. IEEE Transactions on Communication Technology, 15(1), 52\u201360.","journal-title":"IEEE Transactions on Communication Technology"},{"key":"6761_CR27","volume-title":"Fundamentals of statistical signal processing: Estimation theory","author":"SM Kay","year":"1993","unstructured":"Kay, S. M. (1993). Fundamentals of statistical signal processing: Estimation theory. Prentice-Hall Inc."},{"issue":"4","key":"6761_CR28","first-page":"32","volume":"1","author":"A Krizhevsky","year":"2009","unstructured":"Krizhevsky, A., & Hinton, G. (2009). Learning multiple layers of features from tiny images. Handbook of Systemic Autoimmune Diseases, 1(4), 32\u201333.","journal-title":"Handbook of Systemic Autoimmune Diseases"},{"key":"6761_CR29","unstructured":"Langley, P., Iba, W., Thompson, K., et al. (1992). An analysis of bayesian classifiers. In Proceedings of the AAAI conference on artificial intelligence (pp. 223\u2013228)."},{"issue":"11","key":"6761_CR30","doi-asserted-by":"publisher","first-page":"2278","DOI":"10.1109\/5.726791","volume":"86","author":"Y LeCun","year":"1998","unstructured":"LeCun, Y., Bottou, L., Bengio, Y., & Haffner, P. (1998). Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11), 2278\u20132324.","journal-title":"Proceedings of the IEEE"},{"key":"6761_CR31","unstructured":"Li, W., Dasarathy, G., Berisha, V. (2020). Regularization via structural label smoothing. In international conference on artificial intelligence and statistics (pp.1453\u20131463)."},{"issue":"2","key":"6761_CR32","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1109\/TIT.1982.1056489","volume":"28","author":"S Lloyd","year":"1982","unstructured":"Lloyd, S. (1982). Least squares quantization in pcm. IEEE Transactions on Information Theory, 28(2), 129\u2013137.","journal-title":"IEEE Transactions on Information Theory"},{"key":"6761_CR33","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1109\/TPAMI.1987.4767899","volume":"2","author":"G Loizou","year":"1987","unstructured":"Loizou, G., & Maybank, S. J. (1987). The nearest neighbor and the bayes error rates. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2, 254\u2013262.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"6761_CR34","unstructured":"Noshad, M., Xu, L., Hero, A. (2019). Learning to benchmark: Determining best achievable misclassification error from training data. arXiv preprint arXiv:1909.07192"},{"issue":"85","key":"6761_CR35","first-page":"2825","volume":"12","author":"F Pedregosa","year":"2011","unstructured":"Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., & Duchesnay, E. (2011). Scikit-learn: Machine learning in python. Journal of Machine Learning Research, 12(85), 2825\u20132830.","journal-title":"Journal of Machine Learning Research"},{"key":"6761_CR36","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1109\/TASLP.2023.3328283","volume":"32","author":"R Prabhavalkar","year":"2024","unstructured":"Prabhavalkar, R., Hori, T., Sainath, T. N., Schl\u00fcter, R., & Watanabe, S. (2024). End-to-end speech recognition: A survey. IEEE\/ACM Transactions on Audio, Speech, and Language Processing, 32, 325\u2013351.","journal-title":"IEEE\/ACM Transactions on Audio, Speech, and Language Processing"},{"issue":"1","key":"6761_CR37","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/BF00116251","volume":"1","author":"JR Quinlan","year":"1986","unstructured":"Quinlan, J. R. (1986). Induction of decision trees. Machine Learning, 1(1), 81\u2013106.","journal-title":"Machine Learning"},{"key":"6761_CR38","doi-asserted-by":"publisher","first-page":"3793","DOI":"10.1109\/TSP.2020.2994807","volume":"68","author":"SY Sekeh","year":"2020","unstructured":"Sekeh, S. Y., Oselio, B., & Hero, A. O. (2020). Learning to bound the multi-class bayes error. IEEE Transactions on Signal Processing, 68, 3793\u20133807.","journal-title":"IEEE Transactions on Signal Processing"},{"issue":"7","key":"6761_CR39","doi-asserted-by":"publisher","first-page":"4135","DOI":"10.1109\/TIT.2012.2195549","volume":"58","author":"K Sricharan","year":"2012","unstructured":"Sricharan, K., Raich, R., & Hero, A. O. (2012). Estimation of nonlinear functionals of densities with confidence. IEEE Transactions on Information Theory, 58(7), 4135\u20134159.","journal-title":"IEEE Transactions on Information Theory"},{"key":"6761_CR40","unstructured":"Theisen, R., Wang, H., Varshney, L.R., Xiong, C., & Socher, R. (2021). Evaluating state-of-the-art classification models against bayes optimality. In Proceedings of the advances in neural information processing systems (pp. 9367\u20139377)."},{"issue":"10","key":"6761_CR41","doi-asserted-by":"publisher","first-page":"1996","DOI":"10.1038\/s41591-022-01961-6","volume":"28","author":"V Volovici","year":"2022","unstructured":"Volovici, V., Syn, N. L., Ercole, A., Zhao, J. J., & Liu, N. (2022). Steps to avoid overuse and misuse of machine learning in clinical research. Nature Medicine, 28(10), 1996\u20131999.","journal-title":"Nature Medicine"},{"key":"6761_CR42","doi-asserted-by":"crossref","unstructured":"Wang, W., Feng, L., Jiang, Y., Niu, G., Zhang, M-L., & Sugiyama, M. (2024). Binary classification with confidence difference. In Proceedings of the advances in neural information processing systems (pp. 5936\u20135960).","DOI":"10.52202\/075280-0260"},{"issue":"6","key":"6761_CR43","doi-asserted-by":"publisher","first-page":"61003","DOI":"10.1209\/0295-5075\/acc19e","volume":"141","author":"E Xu","year":"2023","unstructured":"Xu, E., Zhou, T., Yu, Z., Sun, Z., & Guo, B. (2023). Equivalence between time series predictability and bayes error rate. Europhysics Letters, 141(6), 61003.","journal-title":"Europhysics Letters"},{"issue":"8","key":"6761_CR44","doi-asserted-by":"publisher","first-page":"1422","DOI":"10.1109\/TKDE.2011.92","volume":"24","author":"S Yang","year":"2012","unstructured":"Yang, S., & Hu, B. (2012). Discriminative feature selection by nonparametric bayes error minimization. IEEE Transactions on Knowledge and Data Engineering, 24(8), 1422\u20131434.","journal-title":"IEEE Transactions on Knowledge and Data Engineering"},{"key":"6761_CR45","unstructured":"Zhou, D., Bousquet, O., Lal, T.N., Weston, J., & Sch\u00f6lkopf, B. (2004). Learning with local and global consistency. In Proceedings of the advances in neural information processing systems pp. (321\u2013328)."}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06761-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-025-06761-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06761-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T00:02:18Z","timestamp":1775865738000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-025-06761-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,11]]},"references-count":44,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["6761"],"URL":"https:\/\/doi.org\/10.1007\/s10994-025-06761-w","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,11]]},"assertion":[{"value":"22 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 January 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 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 declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable. All the experiments in this paper are computer simulations and do not involve experiments on animals, plants, or human entities.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"Not applicable. The paper does not include data or images that require permissions to be published.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}],"article-number":"134"}}