  In frequentist inference, minimizing the Hellinger distance between a kernel
density estimate and a parametric family produces estimators that are both
robust to outliers and statistically efficienty when the parametric model is
correct. This paper seeks to extend these results to the use of nonparametric
Bayesian density estimators within disparity methods. We propose two
estimators: one replaces the kernel density estimator with the expected
posterior density from a random histogram prior; the other induces a posterior
over parameters through the posterior for the random histogram. We show that it
is possible to adapt the mathematical machinery of efficient influence
functions from semiparametric models to demonstrate that both our estimators
are efficient in the sense of achieving the Cramer-Rao lower bound. We further
demonstrate a Bernstein-von-Mises result for our second estimator indicating
that it's posterior is asymptotically Gaussian. In addition, the robustness
properties of classical minimum Hellinger distance estimators continue to hold.
