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MiS Preprint

Finding the Maximizers of the Information Divergence from an Exponential Family

Johannes Rauh


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.

MSC Codes:
94A17, 62B10, 52C40, 13P25
kullback-leibler divergence, relative entropy, exponential family, information projection, optimization, commutative algebra

Related publications

2011 Repository Open Access
Johannes Rauh

Finding the maximizers of the information divergence from an exponential family

In: IEEE transactions on information theory, 57 (2011) 6, pp. 3236-3247