Generalised Pinsker Inequalities

Abstract

We generalise the classical Pinsker inequality which relates variational divergence to Kullback-Liebler divergence in two ways: we consider arbitrary f-divergences in place of KL divergence, and we assume knowledge of a sequence of values of generalised variational divergences. We then develop a best possible inequality for this doubly generalised situation. Specialising our result to the classical case provides a new and tight explicit bound relating KL to variational divergence (solving a problem posed by Vajda some 40 years ago). The solution relies on exploiting a connection between divergences and the Bayes risk of a learning problem via an integral representation.

21 pages, 3 figures, accepted to COLT 2009

References in corpus (2)

• Bounds on Non-Symmetric Divergence Measures in Terms of Symmetric Divergence Measures
• Information, Divergence and Risk for Binary Experiments