{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,21]],"date-time":"2024-08-21T23:38:04Z","timestamp":1724283484881},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"06","license":[{"start":{"date-parts":[[2019,6,13]],"date-time":"2019-06-13T00:00:00Z","timestamp":1560384000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2019,11]]},"abstract":"Abstract<\/jats:title>The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: for a fixed probability measure q<\/jats:italic> on , ( is a finite set), and any probability measure on ,\n(*)<\/jats:label>\n$$D(p||q){\\rm{\\le}}C \\cdot \\sum\\limits_{i = 1}^n {{\\rm{\\mathbb{E}}}_p D(p_i ( \\cdot |Y_1 ,{\\rm{ }}...,{\\rm{ }}Y_{i - 1} ,{\\rm{ }}Y_{i + 1} ,...,{\\rm{ }}Y_n )||q_i ( \\cdot |Y_1 ,{\\rm{ }}...,{\\rm{ }}Y_{i - 1} ,{\\rm{ }}Y_{i + 1} ,{\\rm{ }}...,{\\rm{ }}Y_n )),} $$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula>\nwhere pi<\/jats:sub><\/jats:italic>(\u00b7 |y1<\/jats:sub><\/jats:italic>, \u2026, yi\u22121<\/jats:sub><\/jats:italic>, yi+1<\/jats:sub><\/jats:italic>, \u2026, yn<\/jats:sub><\/jats:italic>) and qi<\/jats:sub><\/jats:italic>(\u00b7 |x1<\/jats:sub><\/jats:italic>, \u2026, xi\u22121<\/jats:sub><\/jats:italic>, xi+1<\/jats:sub><\/jats:italic>, \u2026, xn<\/jats:sub><\/jats:italic>) denote the local specifications for p<\/jats:italic> resp. q<\/jats:italic>, that is, the conditional distributions of the ith coordinate, given the other coordinates. The constant C<\/jats:italic> depends on (the local specifications of) q<\/jats:italic>.<\/jats:p>The inequality (*) ismeaningful in product spaces, in both the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for q<\/jats:italic>, provided uniform logarithmic Sobolev inequalities are available for qi<\/jats:sub><\/jats:italic>(\u00b7 |x1<\/jats:sub><\/jats:italic>, \u2026, xi\u22121<\/jats:sub><\/jats:italic>, xi+1<\/jats:sub><\/jats:italic>, \u2026, xn<\/jats:sub><\/jats:italic>), for all fixed i<\/jats:italic> and fixed (x1<\/jats:sub><\/jats:italic>, \u2026, xi\u22121<\/jats:sub><\/jats:italic>, xi+1<\/jats:sub><\/jats:italic>, \u2026, xn<\/jats:sub><\/jats:italic>). Inequality (*) directly implies that the Gibbs sampler associated with q<\/jats:italic> is a contraction for relative entropy.<\/jats:p>In this paper we derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance.<\/jats:p>","DOI":"10.1017\/s0963548319000099","type":"journal-article","created":{"date-parts":[[2019,6,13]],"date-time":"2019-06-13T07:06:45Z","timestamp":1560409605000},"page":"919-935","source":"Crossref","is-referenced-by-count":17,"title":["Logarithmic Sobolev inequalities in discrete product spaces"],"prefix":"10.1017","volume":"28","author":[{"given":"Katalin","family":"Marton","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,6,13]]},"reference":[{"key":"S0963548319000099_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01019160"},{"key":"S0963548319000099_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2006.10.002"},{"key":"S0963548319000099_ref9","doi-asserted-by":"publisher","DOI":"10.1111\/j.1751-5823.2002.tb00178.x"},{"key":"S0963548319000099_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02101930"},{"key":"S0963548319000099_ref8","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1034968224"},{"key":"S0963548319000099_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02101929"},{"key":"S0963548319000099_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01011153"},{"key":"S0963548319000099_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0096511"},{"key":"S0963548319000099_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-6653-7_21"},{"key":"S0963548319000099_ref11","doi-asserted-by":"publisher","DOI":"10.2307\/2373688"},{"key":"S0963548319000099_ref5","doi-asserted-by":"publisher","DOI":"10.1137\/1115049"},{"key":"S0963548319000099_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/BF00533259"},{"key":"S0963548319000099_ref4","doi-asserted-by":"publisher","DOI":"10.1137\/1113026"},{"key":"S0963548319000099_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/PL00008792"},{"key":"S0963548319000099_ref2","doi-asserted-by":"publisher","DOI":"10.5802\/afst.1460"},{"key":"S0963548319000099_ref1","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199535255.001.0001"},{"key":"S0963548319000099_ref24","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(92)90073-R"},{"key":"S0963548319000099_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02096629"},{"key":"S0963548319000099_ref22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02101094"},{"key":"S0963548319000099_ref20","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2014.2387065"},{"key":"S0963548319000099_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01015569"},{"key":"S0963548319000099_ref19","volume-title":"Une Initiation aux Inegalit\u00e9s de Sobolev Logarithmiques","author":"Royer","year":"1999"},{"key":"S0963548319000099_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2012.10.001"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548319000099","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,7]],"date-time":"2019-10-07T04:58:57Z","timestamp":1570424337000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548319000099\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,13]]},"references-count":23,"journal-issue":{"issue":"06","published-print":{"date-parts":[[2019,11]]}},"alternative-id":["S0963548319000099"],"URL":"http:\/\/dx.doi.org\/10.1017\/s0963548319000099","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,13]]}}}
