Generalised Pinsker Inequalities
Mark Reid and Bob Williamson
In: Conference on Learning Theory 2009, 18-21 June 2009, Montreal.
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.