Finding the Maximizers of the Information Divergence From an Exponential Family

This paper investigates maximizers of the information divergence from an exponential family
${\cal E}$
. It is shown that the
$rI$
-projection of a maximizer
$P$
to
${\cal E}$
is a convex combination of
$P$
and a probability measure
$P_{-}$
with disjoint support and the same value of the sufficient statistics
$A$
. This observation can be used to transform the original problem of maximizing
${D(\cdot \Vert {\cal E})}$
over the set of all probability measures into the maximization of a function
${{\overline D}_{r}}$
over a convex subset of
$\ker A$
. The global maximizers of both problems correspond to each other. Furthermore, finding all local maximizers of
${{\overline D}_{r}}$
yields all local maximizers of
${D(\cdot \Vert {\cal E})}$
.