Avoiding prior–data conflict in regression models via mixture priors

The Bayesian model consists of the prior–likelihood pair. A prior–data conflict arises whenever the prior allocates most of its mass to regions of the parameter space where the likelihood is relatively low. Once a prior–data conflict is diagnosed, what to do next is a hard question to answer. We propose an automatic prior elicitation that involves a two-component mixture of a diffuse and an informative prior distribution that favours the first component if a conflict emerges. Using various examples, we show that these mixture priors can be useful in regression models as a device for regularizing the estimates and retrieving useful inferential conclusions.