MiS Preprint Repository

This paper investigates maximizers of the information divergence from an exponential family $\mathcal{E}$. It is shown that the $rI$-projection of a maximizer $P$ to $\mathcal{E}$ 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 $D(\cdot ||\mathcal{E})$ over the set of all probability measures into the maximization of a function ${\bar{D}}_r$ over a convex subset of $\ker A$. The global maximizers of both problems correspond to each other. Furthermore, finding all local maximizers of ${\bar{D}}_r$ yields all local maximizers of $D(\cdot ||\mathcal{E})$.

This paper also proposes two algorithms to find the maximizers of ${\bar{D}}_r$ and applies them to two examples, where the maximizers of $D(\cdot ||\mathcal{E})$ were not known before.