Bounding posterior means by model criticism

To handle the sensitivity of posterior inferences to prior choices, Leamer (1983) suggests that one specify a class of priors that is large enough to include all possible priors and then derive the set of posterior distributions associated with it. In many practical situations, however, this set of posteriors turns out to be too large to be useful even with informative data. Frequently, the culprit is those dogmatic priors that are not compatible with the data information and removal of such priors often significantly narrows down the set of posteriors. When a complete elicitation process requires excessive time and resources, a feasible alternative is to use the sample information to help the researcher identify unreasonable priors. The approach proposed in this paper is to bound the posterior means of the regression parameters by ‘Model Criticism’ in the sense of Box (1980). More specifically, it restricts priors to those for which the predictive density of the observation is not too small. A set of posteriors is then obtained for this restricted class of priors. The proposed procedure will be applied to U.S. money stock data to examine the behavior of the proposed bounds.