An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces

TL;DR

This paper establishes a new inequality linking relative entropy and logarithmic Sobolev constants in Euclidean spaces, leading to improved inequalities for Gibbs samplers and classical Gaussian cases.

Contribution

It introduces a novel inequality connecting relative entropy with local specifications' Sobolev constants, strengthening existing results for Gaussian distributions.

Findings

Derived a weighted logarithmic Sobolev inequality for Gibbs samplers.

Proved a new inequality relating relative entropy to local specifications.

Extended classical Gaussian Sobolev inequalities to broader contexts.

Abstract

Let $q(x)$ and $p(x)$ denote density functions on the $n$-dimensional Euclidean space, and let $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote their local specifications. For a class of density functions $q$ we prove an inequality between the relative entropy $D(p||q)$ and a weighted sum of the conditional relative entropies $D(p_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) ||Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n))$ that holds for any $p$. The weights are proportional to the logarithmic Sobolev constants of the local specifications of $q$. Thereby we derive a logarithmic Sobolev inequality for a weighted Gibbs sampler governed by the local specifications of $q$. Moreover, this inequality implies a classical logarithmic Sobolev inequality for $q$, as defined for Gaussian distribution by L. Gross. This strengthens a result by F. Otto and M. Reznikoff. The proof is based on ideas developed by F. Otto and C. Villani in their paper on the connection between Talagrand's transportation-cost inequality and logarithmic Sobolev inequality.