Sensitivity in Bayesian Statistics: The Prior and the Likelihood

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Abstract
One paradigm for sensitivity analyses in Bayesian statistics is to specify Γ, a reasonable class of priors, and to compute the corresponding class of posterior inferences. The class Γ is chosen to represent uncertainty about the prior. There is often additional uncertainty, however, about the family of sampling distributions. This article introduces a method for computing ranges of posterior expectations over reasonable classes of sampling distributions that lie “close to” a given parametric family. By treating the prior as a probability measure on the space of sampling distributions this article also gives a unified treatment to what are usually considered two separate problems—sensitivity to the prior and sensitivity to the sampling model. First the notion of “close to” is made explicit. Then, an algorithm is given for turning ratio-linear problems into sequences of linear problems. In addition to solving the problem at hand, the algorithm simplifies many other robust Bayesian computational problems. Finally, the method is illustrated with an example.
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