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

Let $q(x)$ and $p(x)$ denote density functions on the $n$-dimensional Euclidean space, and let $p_i(\cdot \mathrm{\mid }{y}_1,\mathrm{.}\mathrm{.}\mathrm{.},y_{i-1},y_{i+1},\mathrm{.}\mathrm{.}\mathrm{.},y_n)$ and $Q_i(\cdot \mathrm{\mid }{x}_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{i-1},x_{i+1},\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ denote their local specifications. For a class of density functions $q$ we prove an inequality between the relative entropy $D(p\mathrm{\mid }\mathrm{\mid }q)$ and a weighted sum of the conditional relative entropies $D(p_i(\cdot|Y_1,…, Y_{i-1},Y_{i+1},…, Y_n) ||Q_i(\cdot|Y_1,…, Y_{i-1},Y_{i+1},…, Y_n))$thatholdsforany$p$. 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.