  This paper introduces a hierarchical framework to incorporate Hellinger
distance methods into Bayesian analysis. We propose to modify a prior over
non-parametric densities with the exponential of twice the Hellinger distance
between a candidate and a parametric density. By incorporating a prior over the
parameters of the second density, we arrive at a hierarchical model in which a
non-parametric model is placed between parameters and the data. The parameters
of the family can then be estimated as hyperparameters in the model. In
frequentist estimation, minimizing the Hellinger distance between a kernel
density estimate and a parametric family has been shown to produce estimators
that are both robust to outliers and statistically efficient when the
parametric model is correct. In this paper, we demonstrate that the same
results are applicable when a non-parametric Bayes density estimate replaces
the kernel density estimate. We then demonstrate that robustness and efficiency
also hold for the proposed hierarchical model. The finite-sample behavior of
the resulting estimates is investigated by simulation and on real world data.
