Robustness Against Conflicting Prior Information in Regression

Including prior information about model parameters is a fundamental step of any Bayesian statistical analysis. It is viewed positively by some as it allows, among others, to quantitatively incorporate expert opinion about model parameters. It is viewed negatively by others because it sets the stage for subjectivity in statistical analysis. Certainly, it creates problems when the inference is skewed due to a conflict with the data collected. According to the theory of conflict resolution (O'Hagan and Pericchi, 2012), a solution to such problems is to diminish the impact of conflicting prior information, yielding inference consistent with the data. This is typically achieved by using heavy-tailed priors. We study both theoretically and numerically the efficacy of such a solution in a regression framework where the prior information about the coefficients takes the form of a product of density functions with known location and scale parameters. We study functions with regularly varying tails (Student distributions), log-regularly-varying tails (as introduced in Desgagn\'e (2015)), and propose functions with slower tail decays that allow to resolve any conflict that can happen under that regression framework, contrarily to the two previous types of functions. The code to reproduce all numerical experiments is available online.