Entropy under disintegrations

Juan Pablo Vigneaux

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arXiv · 2102.09584

Entropy under disintegrations

Juan Pablo Vigneaux

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Abstract

We consider the differential entropy of probability measures absolutely continuous with respect to a given $\sigma$-finite reference measure on an arbitrary measurable space. We state the asymptotic equipartition property in this general case; the result is part of the folklore but our presentation is to some extent novel. Then we study a general framework under which such entropies satisfy a chain rule: disintegrations of measures. We give an asymptotic interpretation for conditional entropies in this case. Finally, we apply our result to Haar measures in canonical relation.

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Juan Pablo Vigneaux. 2021-02-18. Entropy under disintegrations. https://arxiv.org/abs/2102.09584

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