Bounding relative entropy by the relative entropy of local specifications in product spaces
Abstract
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. Σi=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 i, 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 i. 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.