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arXiv · 0906.1244

Generalised Pinsker Inequalities

Mark D. Reid·Robert C. Williamson
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Abstract

We generalise the classical Pinsker inequality which relates variational divergence to Kullback-Liebler divergence in two ways: we consider arbitrary f-divergences in place of KL divergence, and we assume knowledge of a sequence of values of generalised variational divergences. We then develop a best possible inequality for this doubly generalised situation. Specialising our result to the classical case provides a new and tight explicit bound relating KL to variational divergence (solving a problem posed by Vajda some 40 years ago). The solution relies on exploiting a connection between divergences and the Bayes risk of a learning problem via an integral representation.

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Mark D. Reid, Robert C. Williamson. 2009-06-06. Generalised Pinsker Inequalities. https://arxiv.org/abs/0906.1244

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