Functional inequalities from the fundamental limits of Bayesian estimation

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.