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The aim of this paper is to prove an inequality between relative entropy andthe sum of average conditional relative entropies of the following form: For afixed probability measure $q^n$ on $\mathcal X^n$, ($\mathcal X$ is a finiteset), and any probability measure $p^n=\mathcal L(Y^n)$ on $\mathcal X^n$, wehave \begin{equation}\label{*} D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E_{p^n}D(p_i(\cdot|Y_1,\dots, Y_{i-1},Y_{i+1},\dots, Y_n) || q_i(\cdot|Y_1,\dots,Y_{i-1},Y_{i+1},\dots, Y_n)), \end{equation} where $p_i(\cdot|y_1,\dots,y_{i-1},y_{i+1},\dots, y_n)$ and $q_i(\cdot|x_1,\dots, x_{i-1},x_{i+1},\dots,x_n)$ denote the local specifications for $p^n$ resp. $q^n$. The constant shalldepend on the properties of the local specifications of $q^n$.  Inequality (*) is meaningful in product spaces, both in the discrete and thecontinuous case, and can be used to prove a logarithmic Sobolev inequality for$q^n$, provided uniform logarithmic Sobolev inequalities are available for$q_i(\cdot|x_1,\dots, x_{i-1},x_{i+1},\dots, x_n)$, for all fixed $i$ and allfixed $(x_1,\dots, x_{i-1},x_{i+1},\dots, x_n)$. Inequality (*) directlyimplies that the Gibbs sampler associated with $q^n$ is a contraction forrelative entropy.  We derive inequality (*), and thereby a logarithmic Sobolev inequality, indiscrete product spaces, by proving inequalities for an appropriateWasserstein-like distance. A logarithmic Sobolev inequality is, roughlyspeaking, a contractivity property of relative entropy with respect to someMarkov semigroup. It is much easier to prove contractivity for a distancebetween measures than for relative entropy, since distances satisfy thetriangle inequality, and for them well known linear tools, like estimatesthrough matrix norms can be applied.

Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance