Inference after checking multiple Bayesian models for data conflict and applications to mitigating the influence of rejected priors

• The proposed procedure combines Bayesian model checking with robust Bayes acts to guide inference whether or not the model is found to be inadequate.
• 1. Check each model within a large class of models to determine which models conflict with the data and which are adequate.
• 2. For the set of adequate models, use entropy maximization, other distribution combination, or a robust-Bayes decision rule.
• This proposed procedure is illustrated by the application of a class of hierarchical models to a simple data set.

Two major approaches have developed within Bayesian statistics to address uncertainty in the prior distribution and in the rest of the model. First, methods of model checking, including those assessing prior-data conflict, determine whether the posterior resulting from the model is adequate for purposes of inference and estimation or other decision-making. A potential drawback of this approach is that it provides little guidance for inference in the event that the model is found to be inadequate, that is, in conflict with the data. Second, the robust Bayes approach determines the sensitivity of inferences and decisions to the prior distribution and other model assumptions. This approach includes rules for making decisions on the basis of a set of posterior distributions corresponding to the set of reasonable model assumptions. Drawbacks of the second approach include the inability to criticize the set of models and the lack of guidance for specifying such a set.Those two approaches to model uncertainty are combined into a two-stage procedure in order to overcome each of their limitations. The first stage checks each model within a large class of models to assess which models are in conflict with the data and which are adequate for purposes of data analysis. The resulting set of adequate models is then used in the second stage either for summarizing a combined posterior such as a maximum-entropy posterior or for inference according to decision rules of the robust Bayes approach and of imprecise probability more generally. This proposed framework is illustrated by the application of a class of hierarchical models to a simple data set.