A Robust Conflict Measure of Inconsistencies in Bayesian Hierarchical Models

O'Hagan ("Highly Structured Stochastic Systems", Oxford University Press, Oxford, 2003) introduces some tools for criticism of Bayesian hierarchical models that can be applied at each node of the model, with a view to diagnosing problems of model fit at any point in the model structure. His method relies on computing the posterior median of a conflict index, typically through Markov chain Monte Carlo simulations. We investigate a Gaussian model of one-way analysis of variance, and show that O'Hagan's approach gives unreliable false warning probabilities. We extend and refine the method, especially avoiding double use of data by a data-splitting approach, accompanied by theoretical justifications from a non-trivial special case. Through extensive numerical experiments we show that our method detects model mis-specification about as well as the method of O'Hagan, while retaining the desired false warning probability for data generated from the assumed model. This also holds for Student's-"t" and uniform distribution versions of the model. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..