{
  "abstract": "We develop a Bayesian information-theoretic framework for proving functional inequalities for log-concave measures. Motivated by Klartag and Ordentlich, we establish a unified equivalence between $Φ$-Sobolev inequalities and Bayesian estimation lower bounds: for a large class of smooth convex $Φ$, a log-concave measure $μ$ satisfies a $Φ$-Sobolev inequality with constant $C>0$ if and only if for $X\\simμ$, any $t>0$, and any real-valued measurable $f$, the Bayes risk of estimating $f(X)$ under the $Φ$-Bregman loss from the Gaussian additive observation $Y_t=\\sqrt tX+G, G\\sim N(0,I_n)$ is at least a $1/(1+Ct)$-fraction of that of the trivial estimator. Combining it with direct information-theoretic arguments, such as testing-to-estimation reductions, we obtain old and new functional inequalities: - For the Poincaré inequality, corresponding to $Φ(x)=(x-1)^2$, we give a simple static information-theoretic proof that the Poincaré constant of an isotropic log-concave measure on $\\mathbb{R}^n$ is $O(n^{1/2})$, first proven via stochastic localization in the seminal work of Lee and Vempala. - For the log-Sobolev inequality, corresponding to $Φ(x)=x\\log x$, we prove that all smooth $K$-subgaussian log-concave measures $μ$ satisfy $C_{\\mathrm{LS}}(μ)\\lesssim K^2n^{1/4}$, surpassing the recent state-of-the-art bound of $O(K^2n^{1/2})$ by Bizeul. In parallel and independent work, OpenAI claimed the optimal bound $O(K^2)$ in their October 6, 2026 release of 722 papers. - Finally, we give a Bayesian reformulation of the common structure of the three recent proofs of the Kannan--Lovász--Simonovits conjecture, a dimension-free Poincaré bound for isotropic log-concave measures [BK26; BKL26; SZ26]. Tools from the study of the \"low-degree MMSE\" in computational complexity of statistical inference motivate and recast key steps and quantities of these proofs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05831",
  "author_total": 2,
  "authors": [
    {
      "name": "Shuchen Li"
    },
    {
      "name": "Ilias Zadik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1wzr8e2thd2s4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05831",
  "fulltext_url": null,
  "id": "1wzr8e2thd2s4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05831",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05831",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1wzr8e2thd2s4.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05831",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05831"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Functional inequalities from the fundamental limits of Bayesian estimation",
  "title_text": "Functional inequalities from the fundamental limits of Bayesian estimation",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:32Z",
  "url": "https://research.draconitic.com/paper/1wzr8e2thd2s4",
  "venue": "arXiv",
  "versions": []
}