Local Robustness in Bayesian Analysis
Whereas a global approach to prior robustness focusses on the range of inferences arising from a range of priors, the local approach is concerned with derivatives of posterior quantities with respect to the prior. The local approach has several advantages. First, in contrast to the linearization algorithm used for global robustness, an iterative scheme is not required. Second, it is typically straightforward to obtain a local analysis from Markov chain Monte Carlo posterior output. Third, local sensitivity analysis can be implemented in situations where the linearization algorithm for global analysis is not applicable because the posterior quantity of interest is not a ratio-linear functional of the distribution being perturbed. On the downside, however, it is hard to assess the accuracy of a local sensitivity analysis viewed as an approximation to a global analysis. In particular, summaries of local sensitivity often have nonsensical asymptotic behaviour, raising thorny questions about their calibration. In this article the local approach is broadly defined, ranging from differentiation of functions (e.g., assessing sensitivity to hyperparameters) to differentiation of functionals (e.g., assessing sensitivity to the prior distribution as a whole). In this article we review some of the basic formulations for local assessment of prior influence. We then discuss the use of local analysis to study sensitivity in contexts broader than merely prior uncertainty. Finally, we try to summarize the merits and demerits of local analysis as a robustness tool.