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MiS Preprint
82/2009
Finding the Maximizers of the Information Divergence from an Exponential Family
Johannes Rauh
Abstract
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 $\overline{D}_r$ 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 $D(\cdot||\mathcal{E})$.
This paper also proposes two algorithms to find the maximizers of $\overline{D}_r$ and applies them to two examples, where the maximizers of $D(\cdot||\mathcal{E})$ were not known before.