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

摘要

For a class of density functions q(x) on R ~n we prove an inequality between relative entropy and the weighted sum of conditional relative entropies of the following form: for any density function p(x) on R ~n, where p _i({dot operator}|y _1,..., y _(i-1), y _(i+1),..., y _n) and Q _i({dot operator}|x _1,..., x _(i-1), x _(i+1),..., x _n) denote the local specifications of p respectively q, and ρ _i is the logarithmic Sobolev constant of Q _i({dot operator}|x _1,..., x _(i-1), x _(i+1),..., x _n). 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.