{"paper":{"title":"Geometric Information Decomposition for Weighted Empirical Measures on the Sphere","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"stat.ME","authors_text":"Kisung You","submitted_at":"2026-07-03T04:32:25Z","abstract_excerpt":"We study directional uncertainty when the data already represent a weighted probability measure on the unit sphere, as in importance samples, quadrature rules, or attention-weighted embeddings. A standard approach fits a von Mises-Fisher distribution and reports its concentration or entropy. This is principled but incomplete because vMF uses only mean-direction information and can miss antipodal, axial, girdle-like, or multimodal structure. We introduce the geometric information decomposition (GID), which fits a nested sequence of maximum-entropy projections using spherical features and report"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02943","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02943/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}