{"paper":{"title":"A Lower Bound on the Relative Entropy with Respect to a Symmetric Probability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Matthias Gorny, Rapha\\\"el Cerf","submitted_at":"2014-07-03T09:41:39Z","abstract_excerpt":"Let $\\rho$ and $\\mu$ be two probability measures on $\\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\\mu|\\rho)$ the relative entropy of $\\mu$ with respect to $\\rho$. We prove that, if $\\rho$ is symmetric and $\\mu$ has a finite first moment, then \\[ H(\\mu|\\rho)\\geq \\frac{\\displaystyle{(\\int_{\\mathbb{R}}z\\,d\\mu(z))^2}}{\\displaystyle{2\\int_{\\mathbb{R}}z^2\\,d\\mu(z)}}\\,,\\] with equality if and only if $\\mu=\\rho$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0836","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}