  A number of fundamental quantities in statistical signal processing and
information theory can be expressed as integral functions of two probability
density functions. Such quantities are called density functionals as they map
density functions onto the real line. For example, information divergence
functions measure the dissimilarity between two probability density functions
and are useful in a number of applications. Typically, estimating these
quantities requires complete knowledge of the underlying distribution followed
by multi-dimensional integration. Existing methods make parametric assumptions
about the data distribution or use non-parametric density estimation followed
by high-dimensional integration. In this paper, we propose a new alternative.
We introduce the concept of "data-driven basis functions" - functions of
distributions whose value we can estimate given only samples from the
underlying distributions without requiring distribution fitting or direct
integration. We derive a new data-driven complete basis that is similar to the
deterministic Bernstein polynomial basis and develop two methods for performing
basis expansions of functionals of two distributions. We also show that the new
basis set allows us to approximate functions of distributions as closely as
desired. Finally, we evaluate the methodology by developing data driven
estimators for the Kullback-Leibler divergences and the Hellinger distance and
by constructing empirical estimates of tight bounds on the Bayes error rate.
