On the overall sensitivity of the posterior distribution to its inputs

In a parametric Bayesian analysis, the posterior distribution of the parameter is determined by three inputs: the prior distribution of the parameter, the model distribution of the data given the parameter, and the data themselves. Working in the framework of two particular families of parametric models with conjugate priors, we develop a method for quantifying the local sensitivity of the posterior to simultaneous perturbations of all three inputs. The method uses relative entropy to measure discrepancies between pairs of posterior distributions, model distributions, and prior distributions. It also requires a measure of discrepancy between pairs of data sets. The fundamental sensitivity measure is taken to be the maximum discrepancy between a baseline posterior and a perturbed posterior, given a constraint on the size of the discrepancy between the baseline set of inputs and the perturbed inputs. We also examine the perturbed inputs which attain this maximum sensitivity, to see how influential the prior, model, and data are relative to one another. An empirical study highlights some interesting connections between sensitivity and the extent to which the data conflict with both the prior and the model.