On Prior-Data Conflict in Predictive Bernoulli Inferences

By its capability to deal with the multidimensional nature of uncertainty, imprecise probability provides a powerful methodology to sensibly handle prior-data conflict in Bayesian inference. When there is strong conflict between sample observations and prior knowledge the posterior model should be more imprecise than in the situation of mutual agreement or compatibility. Focusing presentation on the prototypical example of Bernoulli trials, we discuss the ability of different approaches to deal with prior-data conflict. We study a generalized Bayesian setting, including Walley's Imprecise Beta-Binomial model and his extension to handle prior data conflict (called pdc-IBBM here). We investigate alternative shapes of prior parameter sets, chosen in a way that shows improved behaviour in the case of prior-data conflict and their influence on the posterior predictive distribution. Thereafter we present a new approach, consisting of an imprecise weighting of two originally separate inferences, one of which is based on an informative imprecise prior whereas the other one is based on an uninformative imprecise prior. This approach deals with prior-data conflict in a fascinating way.