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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 $\\mu=\\rho$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1407.0836","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-07-03T09:41:39Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"f764112949efb66c6ab2c92ca50551bb2a5351e30427643669d6d588fac366ac","abstract_canon_sha256":"52d346af69212a589308ea4a6153cfe9200ff8bc37d97a9cd6944dbab130bf7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:48.043886Z","signature_b64":"CgY/Suv3YtLfG8gAxLlmL3tVE1K92h4SpIOPmfkIljZDbN9lpW/ievRDP7YsHlNgJEf+PuRGpvFYbfhZNrd9BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f6fac1aa94aa76e8104392129f05c42aef91823a0b825579bc31af4c0994092","last_reissued_at":"2026-05-18T02:39:48.043117Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:48.043117Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $0$. 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