  We propose a geometric framework to assess sensitivity of Bayesian procedures
to modeling assumptions based on the nonparametric Fisher-Rao metric. While the
framework is general in spirit, the focus of this article is restricted to
metric-based diagnosis under two settings: assessing local and global
robustness in Bayesian procedures to perturbations of the likelihood and prior,
and identification of influential observations. The approach is based on the
square-root representation of densities which enables one to compute geodesics
and geodesic distances in analytical form, 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 showcase
the applicability of our framework on several simulated toy datasets as well as
in real data settings for generalized mixed effects models, directional data
and shape data.
