{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/mini-minimax-uncertainty-quantification-for","title":"Mini-Minimax Uncertainty Quantification for Emulators","arxiv_id":"1303.3079","date":"2013-03-13","proceeding":null,"authors":["Jeffrey C. Regier","Philip B. Stark"],"abstract":"Consider approximating a \"black box\" function $f$ by an emulator $\\hat{f}$ based on $n$ noiseless observations of $f$. Let $w$ be a point in the domain of $f$. How big might the error $|\\hat{f}(w) - f(w)|$ be? If $f$ could be arbitrarily rough, this error could be arbitrarily large: we need some constraint on $f$ besides the data. Suppose $f$ is Lipschitz with known constant. We find a lower bound on the number of observations required to ensure that for the best emulator $\\hat{f}$ based on the $n$ data, $|\\hat{f}(w) - f(w)| \\le \\epsilon$. But in general, we will not know whether $f$ is Lipschitz, much less know its Lipschitz constant. Assume optimistically that $f$ is Lipschitz-continuous with the smallest constant consistent with the $n$ data. We find the maximum (over such regular $f$) of $|\\hat{f}(w) - f(w)|$ for the best possible emulator $\\hat{f}$; we call this the \"mini-minimax uncertainty\" at $w$. In reality, $f$ might not be Lipschitz or---if it is---it might not attain its Lipschitz constant on the data. Hence, the mini-minimax uncertainty at $w$ could be much smaller than $|\\hat{f}(w) - f(w)|$. But if the mini-minimax uncertainty is large, then---even if $f$ satisfies the optimistic regularity assumption---$|\\hat{f}(w) - f(w)|$ could be large, no matter how cleverly we choose $\\hat{f}$. For the Community Atmosphere Model, the maximum (over $w$) of the mini-minimax uncertainty based on a set of 1154~observations of $f$ is no smaller than it would be for a single observation of $f$ at the centroid of the 21-dimensional parameter space. We also find lower confidence bounds for quantiles of the mini-minimax uncertainty and its mean over the domain of $f$. For the Community Atmosphere Model, these lower confidence bounds are an appreciable fraction of the maximum.","url_abs":"http://arxiv.org/abs/1303.3079v5","url_pdf":"http://arxiv.org/pdf/1303.3079v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"mini-minimax-uncertainty-quantification-for","repo_url":"https://github.com/jeff-regier/MiniMiniMaxUQ","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}