  We show that for weakly dependent random variables the relative entropy
functional satisfies an approximate version of the standard tensorization
property which holds in the independent case. As a corollary we obtain a family
of dimensionless logarithmic Sobolev inequalities. In the context of spin
systems on a graph, the weak dependence requirements resemble the well known
Dobrushin uniqueness conditions. Our results can be considered as a discrete
counterpart of a recent work of Katalin Marton. We also discuss some natural
generalizations such as approximate Shearer estimates and subadditivity of
entropy.
