Bayesian Linear Regression — Different Conjugate Models and Their (In)Sensitivity to Prior-Data Conflict

The paper is concerned with Bayesian analysis under prior-data conflict, i.e. the situation when observed data are rather unexpected under the prior (and the sample size is not large enough to eliminate the influence of the prior). Two approaches for Bayesian linear regression modeling based on conjugate priors are considered in detail, namely the standard approach also described in Fahrmeir et al. (2007) and an alternative adoption of the general construction procedure for exponential family sampling models. We recognize that – in contrast to some standard i.i.d. models like the scaled normal model and the Beta-Binomial / Dirichlet-Multinomial model, where prior-data conflict is completely ignored – the models may show some reaction to prior-data conflict, however in a rather unspecific way. Finally we briefly sketch the extension to a corresponding imprecise probability model, where, by considering sets of prior distributions instead of a single prior, prior-data conflict can be handled in a very appealing and intuitive way.