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 $\Phi$-Sobolev inequalities and Bayesian estimation lower bounds: for a large class of smooth convex $\Phi$, a log-concave measure $\mu$ satisfies a $\Phi$-Sobolev inequality with constant $C>0$ if and only if for $X\sim \mu$, any $t>0$, and any real-valued measurable $f$, the Bayes risk of estimating $f(X)$ under the $\Phi$-Bregman loss from the Gaussian additive observation $Y_t=\sqrt{t}X+G,G\sim N(0,I_n)$ 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 $\Phi (x)=(x-1{)}^2$, 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 $\Phi (x)=x\log x$, we prove that all smooth $K$-subgaussian log-concave measures $\mu$ satisfy $C_{\mathrm{LS}}(\mu )\lesssim K^2{n}^{1/4}$, surpassing the recent state-of-the-art bound of $O(K^2{n}^{1/2})$ 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.