Reweighted information inequalities

Abstract

We establish a variant of the log-Sobolev and transport-information inequalities for mixture distributions. If a probability measure $\ddot{I}€$ can be decomposed into components that individually satisfy such inequalities, then any measure $\hat{I}¼$ close to $\ddot{I}€$ in relative Fisher information is close in relative entropy or transport distance to a reweighted version of $\ddot{I}€$ with the same mixture components but possibly different weights. This provides a user-friendly interpretation of Fisher information bounds for non-log-concave measures and explains phenomena observed in the analysis of Langevin Monte Carlo for multimodal distributions.