{"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$(t, p_i)$. How accurately can one recover the \"histogram\" (i.e. cumulative density function) of the $p_i$'s? 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02707","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"}