Mixture Data-Dependent Priors

Abstract: We propose a two-component mixture of a noninformative (diffuse) and an informative prior distribution, weighted through the data in such a way to prefer the first component if a prior-data conflict arises. The data-driven approach for computing the mixture weights makes this class data-dependent. Although rarely used with any theoretical motivation, data-dependent priors are often used for different reasons, and their use has been a lot debated over the last decades. However, our approach is justified in terms of Bayesian inference as an approximation of a hierarchical model and as a conditioning on a data statistic. This class of priors turns out to provide less information than an informative prior, perhaps it represents a suitable option for not dominating the inference in presence of small samples. First evidences from simulation studies show that this class could also be a good proposal for reducing mean squared errors.