An Inequality for Relative Entropy and Logarithmic Sobolev Inequalities in Euclidean Spaces

for any density function p(x) on R, where pi(·|y1, . . . , yi−1, yi+1, . . . , yn) and Qi(·|x1, . . . , xi−1, xi+1, . . . , xn) denote the local specifications of p resp. 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 L. Gross. This strengthens a result by F. Otto and M. Reznikoff. The proof is based on ideas developed by F. Otto and C. Villani in their paper on the connection between Talagrand’s transportation-cost inequality and logarithmic Sobolev inequality.