Functional inequalities from the fundamental limits of Bayesian estimation
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∼μ, 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 Yt=√tX+G, G∼ N(0,In) 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 ℝn is O(n1/2), first proven via stochastic localization in the seminal work of Lee and Vempala. - For the log-Sobolev inequality, corresponding to Φ(x)=xlog x, we prove that all smooth K-subgaussian log-concave measures μ satisfy CLS(μ)≲ K2n1/4, surpassing the recent state-of-the-art bound of O(K2n1/2) by Bizeul. In parallel and independent work, OpenAI claimed the optimal bound O(K2) 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.
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BibTeX
@misc{li2026functional,
title = {{Functional inequalities from the fundamental limits of Bayesian estimation}},
author = {Shuchen Li and Ilias Zadik},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05831},
eprint = {2611.05831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05831},
}
RIS
TY - UNPB
TI - Functional inequalities from the fundamental limits of Bayesian estimation
AU - Shuchen Li
AU - Ilias Zadik
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05831
UR - https://arxiv.org/abs/2611.05831
AB - 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∼μ, 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=√tX+G, G∼ 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 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)=xlog x, we prove that all smooth K-subgaussian log-concave measures μ satisfy C_LS(μ)≲ 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.
ER -