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These bounds are post hoc in the sense that the coverage probability holds simultaneously over all <jats:italic>S<\/jats:italic>, possibly chosen depending on the data. This article focuses on the common case of structured null hypotheses, for example, along a tree, a hierarchy, or geometrically (spatially or temporally). Following recent advances in post hoc inference, we build confidence bounds for some prespecified forest\u2010structured subsets and deduce a bound for any subset <jats:italic>S<\/jats:italic> by interpolation. The proposed bounds are shown to improve substantially previous ones when the signal is locally structured. Our findings are supported both by theoretical results and numerical experiments. Moreover, our bounds can be obtained by an algorithm (with complexity bilinear in the sizes of the reference hierarchy and of the selected subset) that is implemented in the open\u2010source <jats:styled-content>R<\/jats:styled-content> package <jats:styled-content>sansSouci<\/jats:styled-content> available from <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/github.com\/pneuvial\/sanssouci\">https:\/\/github.com\/pneuvial\/sanssouci<\/jats:ext-link>, making our approach operational.<\/jats:p>","DOI":"10.1111\/sjos.12453","type":"journal-article","created":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T07:32:40Z","timestamp":1585726360000},"page":"1114-1148","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Post hoc false positive control for structured hypotheses"],"prefix":"10.1111","volume":"47","author":[{"given":"Guillermo","family":"Durand","sequence":"first","affiliation":[{"name":"Laboratoire de probabilit\u00e9s Statistique et Mod\u00e9lisation, LPSM Sorbonne Universit\u00e9  France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gilles","family":"Blanchard","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques d'Orsay Universit\u00e9 Paris\u2010Sud, CNRS, Universit\u00e9 Paris\u2010Saclay  France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierre","family":"Neuvial","sequence":"additional","affiliation":[{"name":"Institut de Math\u00e9matiques de Toulouse UMR 5219, Universit\u00e9 de Toulouse, CNRS, UPS IMT  France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0214-5185","authenticated-orcid":false,"given":"Etienne","family":"Roquain","sequence":"additional","affiliation":[{"name":"Laboratoire de probabilit\u00e9s Statistique et Mod\u00e9lisation, LPSM Sorbonne Universit\u00e9  France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2020,4]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1214\/18-EJS1490"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1214\/18-AOS1721"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1111\/rssb.12028"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1198\/016214504000001907"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1214\/12-AOS1077"},{"key":"e_1_2_8_7_1","volume-title":"Sanssouci: Post hoc inference via joint error rate control. 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