{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T20:59:50Z","timestamp":1776891590876,"version":"3.51.2"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Tsinghua University SIGS Start-up fund","award":["QD2022024C"],"award-info":[{"award-number":["QD2022024C"]}]},{"name":"Tsinghua University SIGS Start-up fund","award":["ZDSYS20220527171406015"],"award-info":[{"award-number":["ZDSYS20220527171406015"]}]},{"name":"Tsinghua University SIGS Start-up fund","award":["JCYJ20220530143002005"],"award-info":[{"award-number":["JCYJ20220530143002005"]}]},{"name":"Shenzhen Ubiquitous Data Enabling Key Lab","award":["QD2022024C"],"award-info":[{"award-number":["QD2022024C"]}]},{"name":"Shenzhen Ubiquitous Data Enabling Key Lab","award":["ZDSYS20220527171406015"],"award-info":[{"award-number":["ZDSYS20220527171406015"]}]},{"name":"Shenzhen Ubiquitous Data Enabling Key Lab","award":["JCYJ20220530143002005"],"award-info":[{"award-number":["JCYJ20220530143002005"]}]},{"name":"Shenzhen Science and Technology Innovation Commission","award":["QD2022024C"],"award-info":[{"award-number":["QD2022024C"]}]},{"name":"Shenzhen Science and Technology Innovation Commission","award":["ZDSYS20220527171406015"],"award-info":[{"award-number":["ZDSYS20220527171406015"]}]},{"name":"Shenzhen Science and Technology Innovation Commission","award":["JCYJ20220530143002005"],"award-info":[{"award-number":["JCYJ20220530143002005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Robustness is an important issue in deep learning, and Bayesian neural networks (BNNs) provide means of robustness analysis, while the minimax method is a conservative choice in the classical Bayesian field. Recently, researchers have applied the closed-loop idea to neural networks via the minimax method and proposed the closed-loop neural networks. In this paper, we study more conservative BNNs with the minimax method, which formulates a two-player game between a deterministic neural network and a sampling stochastic neural network. From this perspective, we reveal the connection between the closed-loop neural and the BNNs. We test the models on some simple data sets and study their robustness under noise perturbation, etc.<\/jats:p>","DOI":"10.3390\/e27040340","type":"journal-article","created":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T06:25:15Z","timestamp":1742883915000},"page":"340","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Minimax Bayesian Neural Networks"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3341-7406","authenticated-orcid":false,"given":"Junping","family":"Hong","sequence":"first","affiliation":[{"name":"Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2608-8034","authenticated-orcid":false,"given":"Ercan Engin","family":"Kuruoglu","sequence":"additional","affiliation":[{"name":"Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,25]]},"reference":[{"key":"ref_1","unstructured":"Bi, K., Xie, L., Zhang, H., Chen, X., Gu, X., and Tian, Q. 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