An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces

For a class of density functions q(x) on Rn we prove an inequality between relative entropy and the weighted sum of conditional relative entropies of the following form: D(p q) Const. n i=1 ρi ·D pi (·|Y1, . . . , Yi−1,Yi+1, . . . , Yn) Qi (·|Y1, . . . , Yi−1,Yi+1, . . . , Yn) for any density function p(x) on Rn, where pi (·|y1, . . . , yi−1, yi+1, . . . , yn) and Qi (·|x1, . . . , xi−1, xi+1, . . . , xn) denote the local specifications of p respectively q, and ρi is the logarithmic Sobolev constant of Qi (·|x1, . . . , xi−1, xi+1, . . . , xn). Thereby we derive a logarithmic Sobolev inequality for a weighted Gibbs sampler governed by the local specifications of q. Moreover, the above inequality implies a classical logarithmic Sobolev inequality for q, as defined for Gaussian distribution by Gross. This strengthens a result by Otto and Reznikoff. The proof is based on ideas developed by Otto and Villani in their paper on the connection between Talagrand’s transportation-cost inequality and logarithmic Sobolev inequality. © 2012 Elsevier Inc. All rights reserved.