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

For a class of density functions q(x)q(x) on RnRn we prove an inequality between relative entropy and the weighted sum of conditional relative entropies of the following form:D(p‖q)⩽Const.∑i=1nρi⋅D(pi(⋅|Y1,…,Yi−1,Yi+1,…,Yn)‖Qi(⋅|Y1,…,Yi−1,Yi+1,…,Yn)) for any density function p(x)p(x) on RnRn, where pi(⋅|y1,…,yi−1,yi+1,…,yn)pi(⋅|y1,…,yi−1,yi+1,…,yn) and Qi(⋅|x1,…,xi−1,xi+1,…,xn)Qi(⋅|x1,…,xi−1,xi+1,…,xn) denote the local specifications of p respectively q  , and ρiρi is the logarithmic Sobolev constant of Qi(⋅|x1,…,xi−1,xi+1,…,xn)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 Talagrandʼs transportation-cost inequality and logarithmic Sobolev inequality.