Sets of priors reflecting prior-data conflict and agreement

Bayesian inference enables combination of observations with prior knowledge in the reasoning process. The choice of a particular prior distribution to represent the available prior knowledge is, however, often debatable, especially when prior knowledge is limited or data are scarce, as then posterior inferences are highly dependent on the choice of prior. Robust Bayesian analysis accounts for this issue by inquiring whether posterior inferences change substantially when the prior distribution is varied within a set of distributions that contains all `reasonable' priors. Similar, but slightly different in scope, is the imprecise probability approach, formalising the idea that sets of probability distributions should be taken to model prior knowledge more accurately. Imprecise probability allows to model prior-data conflict sensitivity: Ranges of posterior inferences should be substantially larger when prior and data are in conflict. Here we propose a new method for generating parameter prior sets in a conjugate setting that, in addition to prior-data conflict sensitivity, allow to reflect strong prior-data agreement, i.e., the case when prior and data coincide especially well, by increased posterior precision. Although presented here for the case of binary data only, it is easily extensible to the general exponential family case