{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,26]],"date-time":"2026-10-26T22:36:35Z","timestamp":1793054195227,"version":"4.5.0"},"reference-count":31,"publisher":"Oxford University Press (OUP)","issue":"4","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"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,10,7]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Watermarking has recently emerged as a crucial tool for protecting the intellectual property of generative models and for distinguishing AI-generated content from human-generated data. Despite its practical success, most existing watermarking schemes are empirically driven and lack a theoretical understanding of the fundamental trade-off between detection power and generation fidelity. To address this gap, we formulate watermarking as a statistical hypothesis-testing problem between a null distribution and its watermarked counterpart. Under explicit constraints on false-positive and false-negative rates, we derive a tight lower bound on the achievable fidelity loss, measured by ageneral $ f $-divergence, and characterize the optimal watermarked distribution that attains this bound. We further develop a corresponding sampling rule that provides an optimal mechanism for inserting watermarks with minimal fidelity distortion. Our result establishes a simple yet broadly applicable principle linking hypothesis testing, information divergence and watermark generation.<\/jats:p>","DOI":"10.1093\/biomet\/asag049","type":"journal-article","created":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T11:15:49Z","timestamp":1786101349000},"source":"Crossref","is-referenced-by-count":0,"title":["Optimal watermark generation under Type I and Type II errors"],"prefix":"10.1093","volume":"113","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-6109-0984","authenticated-orcid":false,"given":"Hengzhi","family":"He","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, University of California, Los Angeles, 8125 Math Sciences Bldg, Los Angeles , California 90095, \u00a0 hengzhihe@g.ucla.edu","place":["U.S.A."]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shirong","family":"Xu","sequence":"additional","affiliation":[{"name":"The Wang Yanan Institute for Studies in Economics, Xiamen University, Economics Building, Xiamen University, Xiamen 361005,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexander","family":"Nemecek","sequence":"additional","affiliation":[{"name":"Department of Computer and Data Sciences, Case Western Reserve University , 10900 Euclid Avenue, Cleveland, Ohio 44106, \u00a0 ajn98@case.edu","place":["U.S.A."]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiping","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Los Angeles , 520 Portola Plaza, MS 6363 Box 951555, Los Angeles , California 90095, U.S.A. jipingli0324@g.ucla.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3383-1081","authenticated-orcid":false,"given":"Erman","family":"Ayday","sequence":"additional","affiliation":[{"name":"Department of Computer and Data Sciences, Case Western Reserve University , 10900 Euclid Avenue, Cleveland, Ohio 44106, \u00a0 exa208@case.edu","place":["U.S.A."]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guang","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, University of California, Los Angeles , 8125 Math Sciences Bldg, Los Angeles , California 90095, \u00a0 guangcheng@ucla.edu","place":["U.S.A."]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,8,8]]},"reference":[{"key":"2026102618184497200_asag049-B1","author":"Aaronson","year":"2022"},{"key":"2026102618184497200_asag049-B2","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1080\/01621459.2026.2618290","article-title":"Towards better statistical understanding of watermarking LLMs","volume":"121","author":"Cai","year":"2026","journal-title":"J. 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