Functional Bregman Divergence and Bayesian Estimation of Distributions

Details Functional Bregman Divergence and Bayesian Estimation of Distributions Functional Bregman Divergence and Bayesian Estimation of Distributions

2006-11-23

arxiv.org/abs/cs/0611123v1

Computer Science

Information Theory

26 pages, 1 figure

Scientific paper

A class of distortions termed functional Bregman divergences is defined, which includes squared error and relative entropy. A functional Bregman divergence acts on functions or distributions, and generalizes the standard Bregman divergence for vectors and a previous pointwise Bregman divergence that was defined for functions. A recently published result showed that the mean minimizes the expected Bregman divergence. The new functional definition enables the extension of this result to the continuous case to show that the mean minimizes the expected functional Bregman divergence over a set of functions or distributions. It is shown how this theorem applies to the Bayesian estimation of distributions. Estimation of the uniform distribution from independent and identically drawn samples is used as a case study.

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