Bounding relative entropy by the relative entropy of local specifications in product spaces

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For a class of density functions qⁿ(xⁿ) on Rⁿ we prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For any density function pⁿ(xⁿ) on Rⁿ, D(pⁿ||qⁿ)≤ Const. ∑ᵢ₌₁ⁿ E D(pᵢ(·|Y₁,..., Yᵢ₋₁,Yᵢ₊₁,..., Yₙ) || Qᵢ(·|Y₁,..., Yᵢ₋₁,Yᵢ₊₁,..., Yₙ)), where pᵢ(·|y₁,..., yᵢ₋₁,yᵢ₊₁,..., yₙ) and Qᵢ(·|x₁,..., xᵢ₋₁,xᵢ₊₁,..., xₙ) denote the local specifications for pⁿ resp. qⁿ, i.e., the conditional density functions of the i'th coordinate, given the other coordinates. The constant depends on the properties of the local specifications of qⁿ. The above inequality implies a logarithmic Sobolev inequality for qⁿ. We get an explicit lower bound for the logarithmic Sobolev constant of qⁿ under the assumptions that: (i) the local specifications of qⁿ satisfy logarithmic Sobolev inequalities with constants ρᵢ, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian of qⁿ are not too large relative to the logarithmic Sobolev constants ρᵢ. Condition (ii) may be weaker than that used in Otto and Reznikoff's recent paper on the estimation of logarithmic Sobolev constants of spin systems.