{"paper":{"title":"The Wasserstein cost of Importance Sampling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Michael Goldman, Simon Coste","submitted_at":"2026-05-28T15:07:45Z","abstract_excerpt":"Importance sampling (IS) consists in biasing samples from a distribution $f$ towards another distribution $g$. Concretely, given samples $X_i$ from $f$, the IS measure is $$\\hat{g}_n = \\frac{1}{Z_n}\\sum_{i=1}^n \\frac{g(X_i)}{f(X_i)} \\delta_{X_i},$$ with $Z_n = \\sum_{i=1}^n \\frac{g(X_i)}{f(X_i)}$. The random measure $\\hat{g}_n$ approximates $g$, and is used in many contexts ranging from Monte Carlo integration to Bayesian inference. We show that, in high dimension ($d \\geqslant 3$), the Wasserstein cost $W_p^p(\\hat{g}_n, g)$ has order $n^{-p/d}$ in expectation, i.e.\n  $$\\beta^{\\mathrm{low}}_{p,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30055","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/2605.30055/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"}