{"paper":{"title":"A uniqueness theorem for the variational free energy decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.IT","math.MP","math.PR"],"primary_cat":"cs.IT","authors_text":"Michael P. Rubin","submitted_at":"2026-07-20T18:57:34Z","abstract_excerpt":"For a finite system with reference measure $p$ and positive weight $\\ell$, the variational free energy $F(Q;p,\\ell)=D(Q\\,\\|\\,p)-E_Q[\\log\\ell]$ satisfies the exact identity $\\log Z(p,\\ell)=-F(Q;p,\\ell)+D(Q\\,\\|\\,\\pi)$, where $Z$ is the partition function and $\\pi$ the associated Gibbs measure. For a uniform reference measure this is the Gibbs-Bogoliubov inequality of mean-field theory; for a Bayesian model it is the evidence decomposition of variational inference. Variational objectives built from $\\alpha$- or R\\'enyi divergences retain useful bounds on $\\log Z$ but not identities of this form, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22710","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/2607.22710/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"}