Geometric Sensitivity Measures for Bayesian Nonparametric Density Estimation Models

TL;DR

This paper introduces a geometric framework using Fisher-Rao metrics to measure the sensitivity of Bayesian nonparametric density estimation models to parameter perturbations, providing new tools for global sensitivity analysis.

Contribution

It develops three novel geometric sensitivity measures based on the Fisher-Rao metric for assessing the impact of hyperparameter changes in Bayesian density models.

Findings

The proposed measures effectively quantify sensitivity in simulated and real data.

Sensitivity measures reveal how hyperparameter perturbations influence posterior densities.

Framework validated through multiple simulation studies and real datasets.

Abstract

We propose a geometric framework to assess global sensitivity in Bayesian nonparametric models for density estimation. We study sensitivity of nonparametric Bayesian models for density estimation, based on Dirichlet-type priors, to perturbations of either the precision parameter or the base probability measure. To quantify the different effects of the perturbations of the parameters and hyperparameters in these models on the posterior, we define three geometrically-motivated global sensitivity measures based on geodesic paths and distances computed under the nonparametric Fisher-Rao Riemannian metric on the space of densities, applied to posterior samples of densities: (1) the Fisher-Rao distance between density averages of posterior samples, (2) the log-ratio of Karcher variances of posterior samples, and (3) the norm of the difference of scaled cumulative eigenvalues of empirical covariance operators obtained from posterior samples. We validate our approach using multiple simulation studies, and consider the problem of sensitivity analysis for Bayesian density estimation models in the context of three real datasets that have previously been studied.