Some aspects of Bayesian robustness

Robustness has always been an important element of the foundation of Statistics. However, it has only been in recent decades that attempts have been made to formalize the problem beyond ad hoc measures towards a theory of robustness. Robustness studies also have recently received considerable interest among Bayesians. Robust Bayesian analysis is the study of the sensitivity of Bayesian answers to uncertain inputs. These uncertain inputs are typically the model, prior distribution, or loss function, or some combination. Based on this reason, in this thesis we study the Bayesian robustness analysis for those typical uncertain inputs using different measures. In addition, the sensitivity analysis or the robustness issues in Bayesian inference can be classified into two broad categories: global and local sensitivity. In this thesis, we will mostly concentrate on the local sensitivity analysis for the prior and the likelihood to investigate the effects of perturbations. For weighted distribution problem, we also consider to perturb the weight function and develop local sensitivity analysis.