Bounding relative entropy by the relative entropy of local specifications in product spaces

TL;DR

This paper establishes an inequality linking the relative entropy of joint distributions to local specifications, leading to explicit bounds on logarithmic Sobolev constants for certain density functions.

Contribution

It introduces a novel inequality connecting global relative entropy to local conditional entropies, enabling explicit bounds on logarithmic Sobolev constants for product space densities.

Findings

Derived an inequality relating relative entropy to local specifications.

Provided explicit lower bounds for the logarithmic Sobolev constant.

Extended previous results by relaxing conditions on mixed partial derivatives.

Abstract

For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For any density function $p^n(x^n)$ on $\Bbb R^n$, $D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E D(p_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) || Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n)),$ where $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote the local specifications for $p^n$ resp. $q^n$, i.e., the conditional density functions of the $i$'th coordinate, given the other coordinates. The constant depends on the properties of the local specifications of $q^n$. The above inequality implies a logarithmic Sobolev inequality for $q^n$. We get an explicit lower bound for the logarithmic Sobolev constant of $q^n$ under the assumptions that: (i) the local specifications of $q^n$ satisfy logarithmic Sobolev inequalities with constants $\rho_i$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian of $q^n$ are not too large relative to the logarithmic Sobolev constants $\rho_i$. Condition (ii) may be weaker than that used in Otto and Reznikoff's recent paper on the estimation of logarithmic Sobolev constants of spin systems.