Robust Bayesian Analysis Given a Bounded Set Probability

摘要

Suppose that just the lower and the upper bounds of the probability of ameasurable subset K in the parameter space Omega are a priori known, when inferences are to be made about measurable subsets A in Omega. Instead of eliciting a unique prior probability measure, consider the class Gamma of all the probability measures compatible with such bounds. Under mild regularity conditions about the likelihood function, the range of the posterior probability of any A is found, as the prior measure varies in Gamma. Such ranges are analyzed according to the robust Bayesian viewpoint. Furthermore, some characterizing properties of the extended likelihood sets are proved. The probability measures in Gamma are then considered as a neighborhood class of an elicited prior, proving that the robust Bayesian interpretation of the likelihood sets, provided by Wasserman (1989), strongly depends on the considered class of priors.