Likelihood Robustness

Most of the research in the theory of Bayesian robustness has concerned the sensitivity of the posterior measures of interest to imprecision solely in the prior, the primary reason being that the prior is perceived to be the weakest link in the Bayesian approach. Another reason is that the operator which maps the likelihood to its posterior is not ratio-linear, making the problem of global robustness with respect to the likelihood, for interesting nonparametric neighborhoods, not very mathematically tractable. Despite these reasons the problem retains its importance, and there have been some interesting studies, which we review here.Initial research has concerned itself with embedding the likelihood in a parametric class or a discrete set of likelihoods; see for example Box and Tiao (1962). Such research suggested methodologies which, though computationally easy, were not considering sufficiently rich neighborhoods of the likelihood. A deviation from this approach is that of Lavine (1991), where the class of neighborhoods chosen was nonparametric. Apart from these, there have been some interesting studies on restricted problems; for instance, imprecision of weights in weighted distributions (Bayarri and Berger, 1998) and the case of regression functions (Lavine, 1994). Until now we were implicitly considering the problem of global robustness, where it is clear that the problem is quite difficult, except for some restricted problems. We conclude by discussing the local sensitivity approach to likelihood robustness where the mathematical problem becomes more tractable (see, e.g., Sivaganesan, 1993), but of course at the cost of ease of interpretation that the global robustness approach entails.