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While the empirical estimates would recover the histogram to earth mover distance $\\Theta(\\frac{1}{\\sqrt{t}})$ (equivalently, $\\ell_1$ distance between the CDFs), we show that, provided $n$ is sufficiently large, we can achieve error $O(\\frac{1}{t})$ which is information theoretical"},"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":"1709.02707","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-08T13:53:26Z","cross_cats_sorted":[],"title_canon_sha256":"7b2d881c818fe254746ccb4695922b51c01ce1ee330ced3a334baa2452937565","abstract_canon_sha256":"5921ce52b5c93244fd118ab6eca7bfa97f6d0ccf3baff818e107e1621a84cc2c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:51.328738Z","signature_b64":"5oxlNgADlLd90Y9a2jGng0FZwa+SGaJYjm5m0zUh9ubCjqZcNI0qjcbh59EwV4cgQ7Jmac9qGtFfd1iSRBsUAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71b30d364f8c2f86343a954924655f3e74dc37d208be6524315a6dedb2b354a8","last_reissued_at":"2026-05-18T00:29:51.328244Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:51.328244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Learning Populations of Parameters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Gregory Valiant, Kevin Tian, Weihao Kong","submitted_at":"2017-09-08T13:53:26Z","abstract_excerpt":"Consider the following estimation problem: there are $n$ entities, each with an unknown parameter $p_i \\in [0,1]$, and we observe $n$ independent random variables, $X_1,\\ldots,X_n$, with $X_i \\sim $ Binomial$(t, p_i)$. 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