Functional inequalities for perturbed measures with applications to log-concave measures and to some Bayesian problems

TL;DR

This paper investigates functional inequalities for perturbed probability measures, providing explicit bounds and applications to log-concave measures and Bayesian statistical methods like Langevin Monte Carlo.

Contribution

It offers new explicit bounds on functional inequality constants for perturbed measures, enhancing their application in Bayesian computation.

Findings

Explicit bounds for Poincaré, Cheeger, and log-Sobolev inequalities.

Applications to Langevin Monte Carlo in Bayesian estimation.

Improved understanding of measure perturbations in statistical contexts.

Abstract

We study functional inequalities (Poincar\'e, Cheeger, log-Sobolev) for probability measures obtained as perturbations. Several explicit results for general measures as well as log-concave distributions are given.The initial goal of this work was to obtain explicit bounds on the constants in view of statistical applications for instance. These results are then applied to the Langevin Monte-Carlo method used in statistics in order to compute Bayesian estimators.