Measuring local sensitivity in Bayesian inference using a new class of metrics

The local sensitivity analysis is recognized for its computational simplicity, and potential use in multi-dimensional and complex problems. Unfortunately, its major drawback is its asymptotic behavior where the prior to posterior convergence in terms of the standard metrics (and also computed by Fréchet derivative) used as a local sensitivity measure is not appropriate. The constructed local sensitivity measures do not converge to zero, and even diverge for the most multidimensional classes of prior distributions. Restricting the classes of priors or using other ϕ-divergence metrics have been proposed as the ways to resolve this issue which were not successful. We overcome this issue, by proposing a new flexible class of metrics so-called credible metrics whose asymptotic behavior is far more promising and no restrictions are required to impose. Using these metrics, the stability of Bayesian inference to the structure of the prior distribution will be then investigated. Under appropriate condition, we present a uniform bound in a sense that a close credible metric a priori will give a close credible metric a posteriori. As a result, we do not get the sort of divergence based on other metrics. We finally show that the posterior predictive distributions are more stable and robust.