  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.
