Abstract
We study functional inequalities (Poincaré, 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. These results are then applied to the Langevin Monte-Carlo method used in statistics in order to compute Bayesian estimators.
Funding Statement
This work has been (partially) supported by the Project EFI ANR-17-CE40-0030 of the French National Research Agency.
Acknowledgements
The authors want to acknowledge an anonymous reviewer for an accurate reading and constructive comments.
Citation
Patrick Cattiaux. Arnaud Guillin. "Functional inequalities for perturbed measures with applications to log-concave measures and to some Bayesian problems." Bernoulli 28 (4) 2294 - 2321, November 2022. https://doi.org/10.3150/21-BEJ1419
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