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Optimally approximating exponential families. (English) Zbl 1283.94027

Summary: This article studies exponential families \(\mathcal E\) on finite sets such that the information divergence \(D(P\|\mathcal E)\) of an arbitrary probability distribution from \(\mathcal E\) is bounded by some constant \(D>0\). A particular class of low-dimensional exponential families that have low values of \(D\) can be obtained from partitions of the state space. The main results concern optimality properties of these partition exponential families. The case where \(D=\log(2)\) is studied in detail. This case is special, because if \(D<\log(2)\), then \(\mathcal{E}\) contains all probability measures with full support.

MSC:

94A15 Information theory (general)
62B10 Statistical aspects of information-theoretic topics
94A17 Measures of information, entropy
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