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

Hellinger Distance and Bayesian Non-Parametrics: Hierarchical Models for Robust and Efficient Bayesian Inference