Generalised Pinsker Inequalities

Understand

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.