Finding the Maximizers of the Information Divergence from an Exponential Family

Johannes Rauh

doi:10.1109/tit.2011.2136230

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arXiv · 0912.4660

Finding the Maximizers of the Information Divergence from an Exponential Family

Johannes Rauh

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Abstract

This paper investigates maximizers of the information divergence from an exponential family $E$. It is shown that the $rI$-projection of a maximizer $P$ to $E$ 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 $D(\cdot||E)$ over the set of all probability measures into the maximization of a function $\Dbar$ over a convex subset of $\ker A$. The global maximizers of both problems correspond to each other. Furthermore, finding all local maximizers of $\Dbar$ yields all local maximizers of $D(\cdot||E)$. This paper also proposes two algorithms to find the maximizers of $\Dbar$ and applies them to two examples, where the maximizers of $D(\cdot||E)$ were not known before.

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Johannes Rauh. 2009-12-23. Finding the Maximizers of the Information Divergence from an Exponential Family. https://doi.org/10.1109/tit.2011.2136230

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