Model checking after Bayesian inference

Bayesian analysis provides a consistent logical framework for processing data, inferring parameters and estimating relevant quantities in engineering prob- lems. However, its outcomes are valid conditional to the specific model assump- tions. Whether these assumptions are questioned, possibly because of some factors knowingly left out, they can be checked by further analysis of the available empirical data. Again, this can be done inside the Bayesian framework, by prob- abilistically comparing expanded models with the original one; however, this may be computational impractical in many applications. Test statistics and p-value analysis, historically developed under the frequentist approach but adapted to the Bayesian setting, provide an alternative for model checking coupled with proba- bilistic inference. In this chapter, we illustrate the relation between p-value analysis and Bayesian model comparison: after presenting it in a general context, we focus on Gaussian linear models under known perturbation, for which this relation can be stated in close formulas, and explore an example outside that domain.