Bayesian sensitivity analysis with the Fisher–Rao metric

We propose a geometric framework to assess sensitivity of Bayesian procedures to modelling assumptions based on the nonparametric Fisher–Rao metric. While the framework is general, the focus of this article is on assessing local and global robustness in Bayesian procedures with respect to perturbations of the likelihood and prior, and on the identification of influential observations. The approach is based on a square-root representation of densities, which enables analytical computation of geodesic paths and distances, facilitating the definition of naturally calibrated local and global discrepancy measures. An important feature of our approach is the definition of a geometric $\epsilon$-contamination class of sampling distributions and priors via intrinsic analysis on the space of probability density functions. We demonstrate the applicability of our framework to generalized mixed-effects models and to directional and shape data.