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

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.

29 pages (in PDF format) Submitted to Journal of Functional Analysis. This paper tackles the same problem as arXiv:0907.4491, but gives a more satisfactory result, and makes arXiv:0907.4491 superfluous