Asymptotic Global Robustness in Bayesian Decision Theory

In Bayesian decision theory, it is known that robustness with respect to the loss and the prior can be improved by adding new observations. In this article we study the rate of robustness improvement with respect to the number of observations n. Three usual measures of posterior global robustness are considered: the (range of the) Bayes actions set derived from a class of loss functions, the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss function ranges over a class. We show that the rate of convergence of the first measure of robustness is √ n, while it is n for the other measures under reasonable assumptions on the class of loss functions. We begin with the study of two particular cases to illustrate our results. 1. Introduction. In Bayesian analysis, choosing a prior distribution and choosing a loss function according to prior knowledge and preferences are difficult tasks. In practice, the decision maker usually chooses convenient approximations to the subjective prior and the subjective loss. The legitimacy of such approximations might be investigated by a sensitivity analysis of the results with respect to the approximations. This is the purpose of robust Bayesian analysis, which recently was overviewed by Ríos Insua and Ruggeri (2000). An interesting approach, called global robustness, proposes to replace a single prior distribution (resp. loss function) by a class of priors (resp. loss functions) and then to compute the range of the ensuing answers as the prior (resp. loss function) varies over the class. Bayesians mainly focus on sensitivity to the prior distribution, although the final result can be drastically affected by the loss function. Moreover, Ru-bin (1987) showed that the loss function and the prior cannot be separated under a weak system of axioms for rational behavior. It is worth pointing