  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.
