Stability and in nitesimal robustness of posterior distributions and posterior quantities

In nitesimal sensitivities of the posterior distribution P(·|X ) and posterior quantities (P) w.r.t. the choice of the prior P are considered. In a very general setting, the posterior P(·|x) and posterior quantities (P) are treated as functions of the prior P on the space M of all probability measures. Qualitative robustness and stability, loosely, then amount to checking if these functions satisfy continuity and Lipschitz condition of order 1. They thus depend on the underlying topology and metric on the space M. It is proved that posterior P(·|X ) and posterior quantity (P) are qualitatively robust in the total variation topology as well as in the weak topology under mild conditions. Qualitative robustness of the Bayes risk, on the other hand, requires rather strong conditions. Stability of posteriors and posterior quantities are also established. An intriguing example shows that simply continuity and boundedness of the likelihood are not enough to guarantee stability of the posterior under the weak convergence metrics. c © 1998 Elsevier Science B.V. All rights reserved. AMS classi cation: 62F15; 62G35; 60B05