  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.
