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

TL;DR

This paper integrates Hellinger distance methods into Bayesian hierarchical models to achieve robust and efficient inference, demonstrating improved performance through simulations and real data analysis.

Contribution

It introduces a hierarchical Bayesian framework incorporating Hellinger distance, extending robustness and efficiency properties to non-parametric Bayesian density estimation.

Findings

Hierarchical model maintains robustness to outliers.

Model achieves statistical efficiency when parametric assumptions hold.

Simulation and real data confirm improved finite-sample performance.

Abstract

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.