Further Results on Geometric Properties of a Family of Relative Entropies

Details Further Results on Geometric Properties of a Family of Relative Entropies Further Results on Geometric Properties of a Family of Relative Entropies

2011-02-24

arxiv.org/abs/1102.4923v4

Computer Science

Information Theory

7 pages, Prop. 5 modified, in Proceedings of the 2011 IEEE International Symposium on Information Theory

Scientific paper

This paper extends some geometric properties of a one-parameter family of relative entropies. These arise as redundancies when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the Kullback-Leibler divergence. They satisfy the Pythagorean property and behave like squared distances. This property, which was known for finite alphabet spaces, is now extended for general measure spaces. Existence of projections onto convex and certain closed sets is also established. Our results may have applications in the R\'enyi entropy maximization rule of statistical physics.

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