Poincaré and Log–Sobolev Inequalities for Mixtures

Entropy

André Schlichting¹ 

[1] Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany;

关键词: Poincaré inequality;    log–Sobolev inequality;    relative entropy;    fisher information;    Dirichlet form;    mixture;    finite Gaussian mixtures;   

DOI  :  10.3390/e21010089

来源: DOAJ

【 摘 要 】

This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the χ 2-distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

【 授权许可】

Unknown