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Let $q(x)$ and $p(x)$ denote density functions on the $n$-dimensionalEuclidean 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 localspecifications. For a class of density functions $q$ we prove an inequalitybetween the relative entropy $D(p||q)$ and a weighted sum of the conditionalrelative 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 $p$. Theweights are proportional to the logarithmic Sobolev constants of the localspecifications of $q$. Thereby we derive a logarithmic Sobolev inequality for aweighted Gibbs sampler governed by the local specifications of $q$. Moreover,this inequality implies a classical logarithmic Sobolev inequality for $q$, asdefined 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'stransportation-cost inequality and logarithmic Sobolev inequality.

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